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Algebra II

Quadratic Functions

www.njctl.org

2014-10-14

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Table of Contents

Explain Characteristics of Quadratic Functions

Graph Quadratic Functions

Solve Quadratic Equations by GraphingSolve Quadratic Equations by Factoring

Solve Quadratic Equations Using Square Roots

Key Terms

Application of Zero Product Property

Solve Quadratic Equations by Completing the SquareSolve Quadratic Equations using the Quadratic Formula

The Discriminant

Combining Transformations (review)

Vertex FormMore Application Problems using Quadratics

click on the topic to go to that section

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Key Terms

Return to Table of Contents

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Zero(s) of a Function: An x value that makes the function equal zero. Also called a "root," "solution" or "x-intercept"

Key Terms

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Minimum Value: The y-value of the vertex if a > 0 and the parabola opens upward

Maximum Value: The y-value of the vertex if a < 0 and the parabola opens downward

Vertex: The highest or lowest point on a parabola.

Key Terms

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Axis of symmetry: The vertical line that divides a parabola into two symmetrical halves

Key Terms

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Explain Characteristics

of Quadratic Functions


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Remember: A quadratic equation is any equation that can be written in the form ax2 + bx + c =0Where a, b, and c are real numbers and a ≠ 0

Question 1: Is a quadratic equation?

Question 2: Is a quadratic equation?

Characteristics of Quadratics

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The form ax2 + bx + c = 0 is called the standard form of a quadratic equation.

The standard form is not unique.

For example,

x2 - x + 1 = 0 can also be written -x2 + x - 1 = 0.

Also, 4x2 - 2x + 2 = 0 can be written 2x2 - x + 1 = 0.

Characteristics of Quadratics

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Practice writing quadratic equations in standard form:(Simplify if possible.)

Write 2x2 = x + 4 in standard form:

Standard Form

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Write 3x = -x2 + 7 in standard form, if possible:

Standard Form

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Write 6x2 - 6x = 12 in standard form and simplify, if possible:

Standard Form

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Write 3x - 2 = 5x in standard form:

Standard Form

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Similar to Quadratic Equations, the standard form of a Quadratic Function is y = ax2 + bx + c, where a ≠ 0.

Notice, a can be positive or negative.

Standard Form

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When graphed, a quadratic function will make the shape of a parabola.

The parabola will open upward if a > 0 or downward if a < 0.

Graph

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The domain of a quadratic function is all real numbers.

Domain

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If the vertex is a maximum, then the range is all real numbers less than or equal to the y-value of the vertex.

The range of this quadratic is

Range

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An axis of symmetry (also known as a line of symmetry) will divide the parabola into mirror images.

The line of symmetry is always a vertical line of the form

Axis of Symmetry

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The x-intercepts are the points at which a parabola intersects the x-axis. These points are also known as zeros, roots or solutions and solution sets. Each quadratic function will have 0, 1, or 2 or real solutions.

2 real solutions no real solutions

1 real solution

X-Intercepts

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1 If a parabola opens downward, the vertex is the highest value on the parabola.

True

False

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2 If a parabola opens upward then...

A a > 0B a < 0C a = 0

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3 The vertical line that divides a parabola into two symmetrical halves is called...

A discriminantB quadratic equationC axis of symmetryD vertexE maximum

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What is the range of the quadratic function below?

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Combining Transformations

(REVIEW)


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Let the graph of f(x) be

Graph y = 2f(.5x+1) - 2


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Graph y =(-1/2 )f(2x + 1) + 2


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Graph y = 3f(-.5x - 2) + 1

Combining TransformationsA

nsw

er

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Graph y = (-1/2)f(-x + 2) +1


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Consider the graph y = x 2 and the rules for stretches and shrinks,

Graph

Graph the Transformation

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7 Given the graph of h(x), which of the following graphs is y = 2h(-x+1) - 3?

A B

C D

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8 Given the graph of h(x), which of the following graphs is y = -0.5h(2x - 1) + 2?

A B

C D

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Graph Quadratic Functions


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Graph by Following Five Steps:

Step 1 - Find Axis of Symmetry

Step 2 - Find Vertex

Step 3 - Find y-intercept

Step 4 - Locate another point

Step 5 - Reflect and Connect

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9 What is the axis of symmetry for y = x 2 + 2x - 3 (Step 1)?

A x = 1B x = -1C x = 2D x = -3

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10 What is the vertex for y = x 2 + 2x - 3 (Step 2)?

A (-1, -4)

B (1, -4)

C (-1, -6)

D (1, -6)

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11 What is the y-intercept for y = x 2 + 2x - 3 (Step 3)?

A (0 , -3)

B (0 , 3)

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Practice: Graph

Graph

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Practice: Graph

Graph

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Practice: Graph

Graph

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Solve Quadratic Equations by

Graphing


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When asked to solve a quadratic equation, there are several ways to do so.

One way to solve a quadratic equation in standard form is to find the zeros of the related function by graphing.

A zero is the point at which the parabola intersects the x-axis.

A quadratic function may have one, two or no zeros.

Solve by Graphing

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How many zeros do the parabolas have? What are the values of the zeros?

No zeroes 2 zeroes; x = -1 and x=3

1 zero;x=1clickclick click

Solve by Graphing

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Every quadratic function has a related quadratic equation.

A quadratic equation is used to find the zeroes of a quadratic function. When a function intersects the x-axis its y-value is zero.

y = ax2 + bx + c Quadratic Function

0 = ax2 + bx + c

ax2 + bx + c = 0 Quadratic Equation

When writing a quadratic function as its related quadratic equation, you replace y with 0.So y = 0.

Vocabulary

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One way to solve a quadratic equation in standard form is to find the zeros or x-intercepts of the related function.

Solve a quadratic equation by graphing:

Step 1 - Write the related function.

Step 2 - Graph the related function.

Step 3 - Find the zeros (or x-intercepts) of the related function.

Solve by Graphing

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2x2 - 18 = 0

2x2 - 18 = y y = 2x2 + 0x - 18

Step 1 - Write the Related Function

Solve by Graphing

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Use the same five-step process for graphing

The axis of symmetry is x = 0.The vertex is (0, -18).The y-intercept is (0, -18).Since the vertex is the y-intercept, locate two other points by substituting values for x. We'll use (2,-10) and (3,0)Graph these points and use reflection across the axis of symmetry. Connect all points with a smooth curve.

Step 2 - Graph the Function

y = 2x2 + 0x – 18

Solve by Graphing

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(3,0) #

x = 0

(2,-10) #

(0,-18)#

#

#

Step 2 - Graph the Function

y = 2x2 + 0x – 18

Solve by Graphing

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The zeros appear to be 3 and -3.

(3,0) #

x = 0

(2,-10) #

(0,-18)#

#

#

Step 3 - Find the zeros

y = 2x2 + 0x – 18

Solve by Graphing

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The zeros are 3 and -3.

Substitute 3 and -3 for x in the quadratic equation.

Check 2x2 – 18 = 0

2(3)2 – 18 = 0 2(9) - 18 = 0 18 - 18 = 0 0 = 0 #

Step 3 - Find the zeros

y = 2x2 + 0x – 18

2(-3)2 – 18 = 0 2(9) - 18 = 0 18 - 18 = 0 0 = 0 #

Solve by Graphing

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12

A

B

C y = -2x2 + 12x - 18

Solve the equation by graphing the related function and identifying the zeros.

-12x + 18 = -2x2

Step 1: Which of these is the related function?

y = 2x2 - 12x - 18

y = -2x2 - 12x + 18

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13 What is the axis of symmetry?

Formula: -b 2a

y = -2x2 + 12x - 18

A x = -3

B x = 3

C x = 4

D x = -5

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14

What is the vertex?

y = -2x2 + 12x - 18

A (3,0)

B (-3,0)

C (4,0)

D (-5,0)

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15

What is the y-intercept?

y = -2x2 + 12x - 18

A (0,0)

B (0, 18)

C (0, -18)

D (0, 12)

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16

A B

C D

If two other points are (5, -8) and (4, -2), what does the graph of y = -2x2 + 12x - 18 look like?

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17

Find the zero(s)

y = -2x2 + 12x - 18

A -18B 4C 3 D -8

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Solve Quadratic Equations by

Factoring


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In addition to graphing, there are additional ways to find the zeros or x-intercepts of a quadratic. This section will explore solving quadratics using the method of factoring.

A complete review of factoring can be found in the Fundamental Skills of Algebra (Supplemental Review) Unit.

Fundamental Skills of Algebra (Supplemental Review)

Click for Link

Solve by Factoring

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Review of factoring - Factoring is simply rewriting an expression in an equivalent form which uses multiplication. To factor a quadratic, ensure that you have the quadratic in standard form: ax2+bx+c=0

Tips for factoring quadratics:· Check for a GCF (Greatest Common Factor). · Check to see if the quadratic is a Difference of Squares or other special binomial product.

Solve by Factoring

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Examples:

Quadratics with a GCF:

3x2 + 6x in factored form is 3x(x + 2)

Quadratics using Difference of Squares:

x2 - 64 in factored form is (x + 8)(x - 8)

Additional Quadratic Trinomials:

x2 - 12x +27 in factored form is (x - 9)(x - 3)2x2 - x - 6 in factored form is (2x + 3)(x - 2)

Solve by Factoring

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https://njctl.org/courses/math/archived-coursesunits-prior-to-common-core/supplemental-units-for-the-transition-to-the-common-core-from-algebra-ii/fundamental-skills-of-algebra/

Practice: To factor a quadratic trinomial of the form x2 + bx + c, find two factors of c whose sum is b.

Example - To factor x2 + 9x + 18, look for factors of 18 whose sum is 9.(In other words, find 2 numbers that multiply to 18 but also add to 9.)

Factors of 18 Sum

Solve by Factoring

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Practice:

Factor x2 + 4x - 12, look for factors of -12 whose sum is 4.(in other words, find 2 numbers that multiply to -12 but also add to 4.)

Factors of -12 Sum

Solve by Factoring

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Zero Product Property

= 0x? ?

Imagine this: If 2 numbers must be placed in the boxes and you know that when you multiply these you get ZERO, what must be true?

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Zero Product Property

For all real numbers a and b, if the product of two quantities equals zero, at least one of the quantities equals zero.

If a b = 0 then a = 0 or b = 0

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Example: Solve x2 + 4x - 12 = 0

1. Factor the trinomial.2. Using the Zero Product Property, set each factor equal to zero.3. Solve each simple equation.

Now... combining the 2 ideas of factoring with the Zero Product Property, we are able to solve for the x-intercepts (zeros) of the quadratic.

Solve by Factoring

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Example: Solve x2 + 36 = 12x

Remember: The equation has to be written in standard form (ax2 + bx + c).

Solve by Factoring

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18 Solve

A x = -7B x = -5C x = -3D x = -2E x = 2

F x = 3

G x = 5

H x = 6

I x = 7

J x = 15

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19 Solve

A m = -7B m = -5C m = -3D m = -2E m = 2

F m = 3

G m = 5

H m = 6

I m = 7

J m = 15

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20 Solve

A h = -12B h = -4C h = -3D h = -2E h = 2

F h = 3G h = 4H h = 6

I h = 8

J h = 12

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21 Solve

A d = -7B d = -5C d = -3D d = -2E d = 0

F d = 3G d = 5H d = 6

I d = 7

J d = 37

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Berry Method to factor

Step 1: Calculate ac.

Step 2: Find a pair of numbers m and n, whose product is ac, and whose sum is b.

Step 3: Create the product .

Step 4: From each binomial in step 3, factor out and discard any common factor. The result is the factored form.

Example: Solve

When a does not equal 1, check first for a GCF, then use the Berry Method.

Berry Method to Factor

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Use the Berry Method.a = 8, b = 2, c = -3

-4 and 6 are factors of -24 that add to +2

Solve

Step 1

Step 2

Step 3

Step 4 Discard common factors


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Use the Zero Product Rule to solve.

Solve


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Solve

Use the Berry Method.a = 4, b = -15, c = -25


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Use the Zero Product Rule to solve.

Solve


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Solve


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Application of the Zero Product Property

In addition to finding the x-intercepts of quadratic equations, the Zero Product Property can also be used to solve real world application problems.


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Example: A garden has a length of (x+7) feet and a width of (x+3) feet. The total area of the garden is 396 sq. ft. Find the width of the garden.

Application

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22 The product of two consecutive even integers is 48. Find the smaller of the two integers.

Hint: Two consecutive integers can be expressed as x and x + 1. Two consecutive even integers can be expressed as x and x + 2.

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23 The width of a rectangular swimming pool is 10 feet less than its length. The surface area of the pool is 600 square feet. What is the pool's width?

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24 A science class designed a ball launcher and tested it by shooting a tennis ball straight up from the top of a 15-story building. They determined that the motion of the ball could be described by the function: , where t represents the time the ball is in the air in seconds and h(t) represents the height, in feet, of the ball above the ground at time t. What is the maximum height of the ball? At what time will the ball hit the ground? Find all key features and graph the function. Students type their answers here

Problem is from:

Click link for exact lesson.

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https://www.engageny.org/resource/algebra-i-module-4-topic-lesson-9



25 A ball is thrown upward from the surface of Mars with an initial velocity of 60 ft/sec. What is the ball's maximum height above the surface before it starts falling back to the surface? Graph the function. The equation for "projectile motion" on Mars is:

Students type their answers here

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Solve Quadratic Equations Using

Square Roots


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What if x2 has a coefficient other than 1?

Example: Solve 4x2 = 20 using the square roots method.

Solve Using Square Roots

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26 When you take the square root of a real number,

your answer will always be positive.

True

False

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27 If x2 = 16, then x =

A 4B 2C -2D 26E -4

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28 Solve using the square root method.

A 5B 20C 4D -2

E -5F 2G -4H -20

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29 If y2 = 4, then y =

A 4B 2C -2D 26E -4

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32 If (3g - 9)2 + 7= 43, then g =

A

B

C

D

E

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Challenge: Solve (2x - 1)² = 20 using the square root method.

Solve Using Square Roots

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33 A physics teacher put a ball at the top of a ramp and let it roll toward the floor. The class determined that the height of the ball could be represented by the equation, ,where the height, h, is measured in feet from the ground and time, t, in seconds. Determine the time it takes the ball to reach the floor.


Problem is from:


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34 A rock is dropped from a 1000 foot tower. The height of the rock as a function of time can be modeled by the equation: . How long does it take for the rock to reach the ground?


Ans

wer

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Solving Quadratic Equations by

Completing the Square


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In algebra, "Completing the Square" is a technique for changing a quadratic expression from standard form: ax2 + bx + c to the vertex/graphing form: a(x + h)2 + k.

It can also be used as a method for solving quadratic equations.


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35 Find (b/2)2 if b = 14.

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36 Find (b/2)2 if b = 10.

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37 Find (b/2)2 if b = -12.

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38 Complete the square to form a perfect square trinomial.

x2 + 18x + ?

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39 Complete the square to form a perfect square trinomial.

x2 - 6x + ?

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Step 1 - Write the equation in the form x2 + bx = c.

Step 2 - Find (b ÷ 2)2.

Step 3 - Complete the square by adding (b ÷ 2)2 to both sides of the equation.

Step 4 - Factor the perfect square trinomial.

Step 5 - Take the square root of both sides.

Step 6 - Write two equations, using both the positive and negative square root and solve each equation.


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Let's look at an example to solve:x2 + 14x -15 = 0

Step 1 - Rewrite Equation

Step 2 - Find (b/2)2

Step 3 - Add the result to both sides

Step 4 - Factor & Simplify

Step 5 - Take Square Root of both sides

Step 6 - Write 2 Equations & Solve

How can you check your solutions?


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Let's look at an example to solve:







How can you check your solutions?


x2 - 2x - 2 = 0

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40 Solve the following by completing the square :

x2 + 6x = -5

A -5B -2C -1D 5E 2

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x2 - 8x = 20

A -10B -2C -1D 10E 2

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Challenge:








3x2 - 10x = -3 *Note: There is no GCF to factor out like the previous example.

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Challenge:








4x2 - 17x + 4 = 0 *Note: There is no GCF to factor out.

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Challenge:Completing the Square

-6x2 - 25x - 25 = 0 *Note: There is no GCF to factor out.

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A

B

C

D

E

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Solve Quadratic Equations by Using

the Quadratic Formula

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At this point you have learned how to solve quadratic equations by:

Today we will be given a tool to solve ANY quadratic equation, and it ALWAYS works!

Many quadratic equations may be solved using these methods. Though completing the square works for any quadratic equation, it can be cumbersome to repeatedly use the algorithm.

· graphing· factoring· using square roots and · completing the square

Solving Quadratics

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Now try Completing the Square on the standard form of a quadratic equation.








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Steps 2 and 3 - Find (b/2)2 , Add the result to both sides, simplify

Step 1 - Rewrite Equation and factor out a



Step 6 - Solve for x


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Example 1: Solve 2x2 + 3x - 5 = 0

a = 2 b = 3 c = -5

Once you identify the values of a, b, and c, simply substitute into the quadratic formula and simplify as much as possible.

How can you check your answers?

Quadratic Formula

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Example 2: Solve: 2x = x2 - 3

To use the Quadratic Formula, the equation must be in standard form (ax2 + bx +c = 0).

Identify a, b, and c, then substitue into the formula and simplify.

Don't forget to check your results!

Quadratic Formula

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44 Solve the following equation using the quadratic formula:

A -5B -4C -3D -2E -1

F 1

G 2

H 3

I 4J 5

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A -5B -4C -3D -2E -1

F 1

G 2

H 3

I 4J 5

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A -5B -4C -3D -2E -1.5

F 1.5

G 2

H 3

I 4J 5

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Example 3: Solve using the quadratic formula, and simplify the result. x2 - 2x - 4 = 0

Quadratic Formula

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47 Find the larger solution to the equation.

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48 Find the smaller solution to the equation.

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Work in small groups to solve the quadratic equation using the following different methods.

Factoring Quadratic Formula


Graphing

Which Method

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Factoring Quadratic Formula


Graphing

Work in small groups to solve the quadratic equation using the following different methods.

Which Method

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The Discriminant


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2 real solutions no real solutions

1 real solution

Recall what it means to have 0, 1, or 2 solutions/zeros/roots

Solutions

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At times, it is not necessary to solve for the zeros or roots of a quadratic function, but simply to know how many roots exist (zero, one, or two).

The quickest way to determine how many solutions a quadratic has, algebraically, is to calculate what's called the discriminant.

It may look familiar, as the discriminant is a part of the quadratic formula.

The Discriminant

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Other important tips before practice:

· The square root of a positive number has two solutions.

· The square root of zero is 0.

· The square root of a negative number has no real solution.

The Discriminant

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If b2 - 4ac > 0 (POSITIVE) the quadratic has two real solutions

If b2 - 4ac = 0 (ZERO) the quadratic has one real solution

If b2 - 4ac < 0 (NEGATIVE) the quadratic has no real solutions

CONCLUSION:

The Discriminant

click to reveal

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49 What is the value of the discriminant of 2x2 - 2x + 3 = 0 ?

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50 Use the discriminant to find the number of solutions for 2x2 - 2x + 3 = 0

A 0

B 1

C 2

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51 What is the value of the discriminant of x2 - 8x + 4 = 0 ?

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52 Use the discriminant to find the number of solutions for x2 - 8x + 4 = 0

A 0

B 1

C 2

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Vertex Form


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Vertex Form

A quadratic equation in vertex form:

So far, we have been using quadratics in standard form. However, sometimes when graphing, it is more useful to write them in Vertex Form.

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Vertex Form shows the location of the vertex (h , k).The a still tells the direction of opening.And the axis of symmetry is x = h.

Example: Find the vertex, direction of opening and the axis of symmetry for the graph of:

A quadratic function written in vertex form:

Vertex Form

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Find the vertex, direction of openness and the axis of symmetry for each.

A.

B.

C.

Vertex Form

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D.

E.

Vertex FormFind the vertex, direction of openness and the axis of symmetry for each.

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53 Find the vertex for the graph of

A

B

C

D

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54 Find the direction of opening for the graph of

A up

B down

C left

D right

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57 Give the direction of opening for the graph of

A up

B down

C left

D right

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58 Give the axis of symmetry for the graph of

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60 Give the direction of openness of

A up

B down

C left

D right

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61 The axis of symmetry for the graph of is ______ .

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Converting from Standard Form to Vertex Form

To convert from standard form to vertex form, we need to recall the method for completing the square.

Step 1 - Write the equation in the form y = x2 + bx + ___ + c - ___

Step 2 - Find (b ÷ 2)2

Step 3 - Write the result from Step 2 in the first blank and in the second blank.

Step 4 - Rewrite the first three terms as a perfect square.

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65 What is the vertex form of:

A

B

C

D

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Compare the following functions based on information from the equations. What do the graphs have in common? How are they different? Sketch both graphs to confirm your conclusions.

Comparing Functions

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Write two different quadratic equations whose graphs have vertices at (3.5, -7).

Two Functions

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What if "a" does not equal 1?

Step 1 - Write the equation in the form y = ax2 + bx +__+ c - __

Step 2 - Factor: y = a(x2 + (b/a)x +__)+ c - __

Step 3 - Find (b/a ÷ 2)2

Step 4 - Put your answer from Step 3 in the first blank and multiply Step 3 by a to fill in the second blank.

Step 5 - Write trinomial as perfect square.

Standard Form to Vertex Form

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Geometric Definition of a Parabola

A parabola is a locus* of points equidistant from a fixed point, the focus, and a fixed line, the directrix.

*locus is just a fancy word for set.

Every parabola is symmetric with respect to a line through the focus and perpendicular to the directrix. The vertex of the parabola is the "turning point" and is on the axis of symmetry.

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Axis of Symmetry

Directrix

Focus

L1

L2

L1=L2

Focus and Directrix of a Parabola

Every point on the parabola is the same distance from the directrix and the focus.

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ParabolasThe parts are the same for all parabolas,regardless of the direction

in which they open.

Directrix

Axis of Symmetry

Vertex

Focus

y=ax2+bx+c

Vertex

Focus

Directrix

Axis of Symmetry

x=ay2+by+c

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Genera l Form y= ax2 + bx + c x= ay2 +by + c

Vertex Form y= a(x - h)2 +k x= a (y - k)2 + h

Opens a>0 opens upa<0 opens down

a>0 opens to the righta<0 opens to the le ft

Axis of Symmetry x = h y = k

Vertex (h , k) (h , k)

Directrix

Focus

Parabola Summary

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71 What is the vertex of ?

A (-3, 2)

B (-3, -2)

C (2, 3)

D (-2, -3)

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72 What is the vertex of ?

A (3, 2)

B (-3, -2)

C (2, 3)

D (-2, -3)

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78 What is the vertex of y= x2 - 8x +21?

A (4, 5)

B (-4, 5)

C (-5, 4)

D (5, 4)

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79 What is the equation of the directrix for the following equation?

A y = 2B y = -4

C x = 3

D x = -5

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80 Where is the focus for the following equation?

A (-3 , 5)B (3 , 5)

C (5 , 3)

D (5 , -3)

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82 What is the equation of the parabola with vertex (2,3) and directrix y = 4?

A y = 4(x - 2)2 + 3

y = -1/4(x - 2)2 + 3

x = 4(y - 2)2 + 3

x = 1/4(y - 2)2 + 3

B

C

D

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More Application Problems Using

Quadratics


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Quadratic Functions in the Real WorldClick on the bike to learn more.

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page1svg

http://njc.tl/12n

Quadratic Equations and ApplicationsA sampling of applied problems that lend themselves to being solved by quadratic equations:

Number Reasoning

Free Falling Objects

Geometry: Dimensions

Distances

Business:Interest Rate

Height of a Projectile

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PLEASE KEEP THIS IN MIND.

When solving applied problems that lead to quadratic equations, you might get a solution that does not satisfy the physical constraints of the problem.

For example, if x represents a width of a garden and the two solutions of the quadratic equations are -9 and 1, the value -9 is rejected since a width must be a positive number. We call this an extraneous solution.

Quadratic Equations and Applications

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The product of two consecutive negative integers is 1122. What are the numbers?

Applications

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Applications

Two cars left an intersection at the same time, one heading north and one heading west. Some time later, they were exactly 100 miles apart. The head headed north had gone 20 miles farther than the car headed west. How far had each car traveled?

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The product of two consecutive odd integers is 1 less than four times their sum. Find the two integers.

Applications

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The length of a rectangle is 6 inches more than its width. The area of the rectangle is 91 square inches. Find the dimensions of the rectangle.

Applications

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83 The product of a number and a number 3 more than the original is 418. What is the smallest value the original number can be?

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84 The product of two consecutive positive even integers is 168. Find the larger of the numbers.

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85 Two cars left an intersection at the same time, one heading north and the other heading east. Some time later they were 200 miles apart. If the car heading east traveled 40 miles farther, how far did the northbound car go?

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86 A square's length is increased by 6 units and its width is increased by 4 units. The result of this transformation is a rectangle with an area of 195 square units. Find the area of the original square.

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87 In the accompanying diagram, the width of the inner rectangle is represented by x - 3 and its length by x + 3. The width of the outer rectangle is represented by 3x + 4 and the length by 3x - 4. Express the area of the pink shaded region as a polynomial in terms of x.


Problem is from:


Step 1: Write an expression to represent the area of the larger rectangle.Step 2: Write an expression to represent the area of the smaller rectangle.Step 3: Subtract the polynomial to get your final answer.

Use the the next page for space to solve.

Application ProblemsApplication Problems

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Step 1: Write an expression to represent the area of the larger rectangle.

Step 2: Write an expression to represent the area of the smaller rectangle.

Step 3: Subtract the polynomial to get your final answer.

Ans

wer

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88 A large painting in the style of Rubens is 3 ft. longer than it is wide. If the wooden frame is 12 in. wide, the area of the picture and frame is 208 ft2 , find the dimensions of the painting. (Draw a diagram.)


Ans

wer

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89 The rectangular picture frame below is the same width all the way around. The photo it surrounds measures 17" by 11". The area of the frame and photo combined is 315 sq. in. What is the length of the outer frame?

17 x x 11

x

x Ans

wer

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90 Two mathematicians are neighbors. Each owns a separate rectangular plot of land that shares a boundary and have the same dimensions. They agree that each has an area of square units. One mathematician sells his plot to the other. The other wants to put a fence around the perimeter of his new combined plot of land. How many linear units of fencing will he need? Write your answer as an expression of x. Students type their answers here

Problem is from:


Note: This question has two correct approaches and two different

correct solutions. Can you find them both? Hint: Start by factoring.

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91 Part A

An expression is given. x2 - 8x + 21 Determine the values of h and k that make the expression (x - h)2 + k equivalent to the given expression.

Input your answer for h =

From PARCC sample test

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92 Part A

An expression is given. x2 - 8x + 21 Determine the values of h and k that make the expression (x - h)2 + k equivalent to the given expression.

Input your answer for k =


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93 Part B

An equation is given.

x2 - 8x + 21 = (x - 4)2 + 3x - 16

Find the one value of x that is a solution to the given equation.


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http://practice.parcc.testnav.com/#



slide 2 / 222 algebra ii

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