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Name:______________ Block:______________

Unit 4 Part 1: Graphing Quadratic Functions

Day 1: Vertex Form

Day 2: Intercept Form

Day 3: Standard Form

Day 4: Review Day 5: Quiz

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Quadratic Functions Day1: Introducing ….. the QUADRATIC function

We will: learn a new function family I will:

Vertex Form: y = a(x – h) 2 + k

Welcome to our second function family …… the QUADRATIC FUNCTION f(x) = x2 (the parent function) Remind you of anything? What are some characteristics that you notice? What is different between this function and the absolute value function? Why? (look at the table…)

Vertex form of a quadratic function: f(x) = a(x – h)2 + k

ALL quadratic functions have key features that we care about:

1. Vertex

2. Axis of symmetry

3. Min or max

4. X-intercepts

5. Y-intercepts

6. Increasing and Decreasing Intervals

7. End behavior

8. Domain

9. Range

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Parent Function: y = x2 Vertex Form: y = a (x – h)2 + k

1. Vertex: ( , )

x y

2. y 2(x 3)2 8

Vertex: ( , ) Table

TURN AND TALK: Make some predictions…..

Given the quadratic function y = a (x - h)2 + k * “a” is NOT slope in other functions! **

If a > 0, does the graph open up or down? _________

If a < 0, does the graph open up or down? _________

If |a| > 1, does the graph get taller and more narrow or shorter and wider?________

If 0 < |a| < 1, does the graph get taller and more narrow or shorter and wider?________

What is the vertex? _________

What does the parameter k control? _________________________

What does the parameter h control? _________________________

Write an equation of a quadratic function with a vertex at (-2, 5) that opens down and has a

dilation.

TURN AND TALK: Look at your quadratic. If you were simplifying it, which would you do first?

Square the binomial or distribute your dilation factor? WHY?

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How do you graph without a calculator? 1. Find your vertex.

2. Place your vertex in the middle of the table of values.

3. Fill in the x-values that surround the vertex.

4. Plug in x-values to find the y-values for your remaining points.

3. y = -2x2

X Y

4. y = ½x2 X Y

5. y = (x - 5)2 - 7

X Y

6. y = - (x + 2)2 + 4

X Y

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Let’s review all the characteristics of our graphs and how to find an inverse of a given function…. 7. Graph: y = –2(x + 2)2 + 2 8. Is the inverse of #7 a function? ____ Explain: Graph the inverse of #7.

Domain: _____________________ Domain: _____________________

Range: _______________________ Range: _______________________

Increasing: ____________________ Increasing: ____________________

Decreasing: ___________________ Decreasing: ___________________

Zeroes: _______________________ Zeroes: _______________________

Y-intercept: ___________________ Y-intercept: ___________________

As

x, f (x )____ As

x, f (x )____

As x, f (x )____ As

x, f (x )____

Complete the table below:

TURN AND TALK: How can you tell if a vertex is a max or min without graphing?

Function Direction Dilation Or

Standard

Vertex Domain Range

1 y = - 2 (x + 6)2 + 3

Up Down

2 y = –

12

(x – 4)2 + 5

Up Down

3 y 4(x 1)2 2

Up Down

4 2 1y x

Up Down

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So what about "a"? How did we find the dilation factor for absolute value? Why can't we do that for quadratic functions? Find the equation of a graphed function given the vertex: Step 1: Find general form of parent graph. Step 2: Find (h, k) and another point on graph (x, y). Step 3: Substitute values into the parent equation and solve for “a”. Step 4: Write the equation. Given the graph, write the quadratic equation for each of the following: 9. 10.

The vertex is __________ . The vertex is __________ .

Equation:_____________________ Equation:_____________________

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ALWAYS, SOMETIMES, NEVER?

Tell whether each statement is always, sometimes, or never true.

1. The graph of a quadratic function is a “V” shape.

2. The range of a quadratic function is the set of all real numbers.

3. The highest power in a quadratic function is 2.

4. The graph of a quadratic function contains the point (0, 0).

5. The vertex of a parabola occurs at the minimum value of the function.

6. The graph of a quadratic function that has a minimum opens upward.

7. The graphs of 𝑓(𝑥) = 𝑎𝑥2 and 𝑔(𝑥) = −𝑎𝑥2 have the same width.

8. A quadratic function has two real solutions.

9. A quadratic function that has its vertex on the x-axis has exactly one solution.

10. A quadratic that function opens down has no solutions.

11. A quadratic function has an axis of symmetry.

12. The inverse of a quadratic function is also a function.

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Quadratic Function Day2: Intercept Form We will: investigate a different form of a quadratic function I will:

y = a(x – p)(x – q)

Is this really a quadratic? Graph these with your calculator and see.

1. y = (x + 3)(x - 1) (p = ____, q = _____) Verify algebraically:

2. y = 2(x - 1)(x - 4) (p = ____, q = _____)

Verify algebraically:

3. f(x) = ½(x + 2)(x - 2) (p = ____, q = _____)

Verify algebraically: What patterns do you notice? TURN AND TALK: What is NICE about this form? What is NICE about VERTEX form?

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How will we find the vertex and axis of symmetry given this form?

4. y = (x – 2)(x – 6) 5. f(x) = -½(x + 6)(x – 2)

x-intercepts: ______, ______ x-intercepts: ______, ______

Vertex: ______ Vertex: ______

Domain: _______ Domain: _______

Range: _________ Range: _________

y-intercept: ________ y-intercept: ________

Increasing Interval: Increasing Interval:

Decreasing Interval: Decreasing Interval:

Max or min? Max or min?

As x, f (x )____ As

x, f (x )____

As x, f (x )____ As

x, f (x )____

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Graphing in intercept form:

1. Find & graph the X-intercepts.

2. Find & graph the vertex. 3. Connect the points to

make the parabola.

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Sketch the graph of a quadratic function that has at least one solution of x=0. 6. y = -3x(x – 2) 7. Y = (x – 4)(x + 2)

x-intercepts: ______, ______ x-intercepts: ______, ______

Vertex: ______ Vertex: ______

Domain: _______ Domain: _______

Range: _________ Range: _________

y-intercept: ________ y-intercept: ________

Increasing Interval: Increasing Interval:

Decreasing Interval: Decreasing Interval:

Max or min? Max or min?

As x, f (x )____ As

x, f (x )____

As x, f (x )____ As

x, f (x )____

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8. What happens if we give #7 a dilation factor? New quadratic in vertex form _______________________ Does "a" affect the intercepts? 9. How would you graph the following function?

y = (x – 3)2

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Quadratic Functions Day3: Standard Form

We will: learn yet ANOTHER form of the quadratic function I will:

y = ax2 + bx + c What was AWESOME about the VERTEX form of a quadratic? What was AWESOME about the INTERCEPT form of a quadratic? Do you see any helpful information in the STANDARD FORM of a quadratic? What will be a little bit more challenging?

Standard Form: y = ax2 + bx + c Summary of STANDARD FORM

Vertex has x-coordinate _______. (How will you know if this is a min or a max?)

Find the y-coordinate of the vertex by plugging the x value of the vertex into the equation.

The vertex is the ordered pair

, ( )2 2

b bf

a a.

The axis of symmetry is x = _______

What happens at the y-intercept?

Then the y-intercept is ____. So, the point (0, ___) is on the parabola.

If a is positive, ______________________.

If a is negative, ______________________.

The solutions to the quadratic equation are the x-intercepts. What can we do to find these

when we are given standard form?

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Steps for Graphing: y = x2 – 2x – 8 y = –2x2 – 8x + 1

Step 1: Find the vertex: (x, y)

Formula: x

b

2a

Plug x into the function to find y.

Step 2: Complete a table of values

Place Vertex in middle.

Fill in x-values.

Pick x-values on one side of vertex to plug in.

Use symmetry to fill in the remaining values.

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Step 3: Graph your points and connect.

y-intercept: ( , )

x-intercept: ( , )

Axis of Symmetry

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Let’s review all the characteristics of our graphs …

3. y = 2x2 + 4x + 5 4. y = 1

3x2 + 2x

vertex: vertex:

Domain: _____________________ Domain: _____________________

Range: _______________________ Range: _______________________

Increasing: ____________________ Increasing: ____________________

Decreasing: ___________________ Decreasing: ___________________

Zeroes: _______________________ Zeroes: _______________________

Y-intercept: ___________________ Y-intercept: ___________________

As x, f (x )____ As

x, f (x )____

As x, f (x )____ As

x, f (x )____

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