introduction imagine the path of a basketball as it leaves a player’s hand and swooshes through...

IntroductionImagine the path of a basketball as it leaves a player’s hand and swooshes through the net. Or, imagine the path of an Olympic diver as she arcs away from the diving board, slips deep into the water, and resurfaces. Both traveling paths make a U-shaped curve known as a parabola. A parabola is the U-shaped graph of a quadratic function, which is an equation with a degree of 2. Formally, a quadratic function is a function that can be written in the form f(x) = ax2 + bx + c, where a ≠ 0. Quadratics can be used to model varying situations. In this lesson, you will learn about and practice identifying the important points on a parabola and use those points to sketch the corresponding graph. 1

5.3.1: Creating and Graphing Equations Using Standard Form

Key Concepts• The standard form of a quadratic function is f(x) =

ax2 + bx + c, where a is the coefficient of the quadratic term, b is the coefficient of the linear term, and c is the constant term.

• The x-intercepts of the graph of a quadratic function are the points at which the graph crosses the x-axis, and are written as (x, 0).

• The y-intercept of the graph of a quadratic function is the point at which the graph crosses the y-axis and is written as (0, y).

• The graph on the following slide shows the location of the parabola’s y-intercept. 2

5.3.1: Creating and Graphing Equations Using Standard Form

Key Concepts, continued

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5.3.1: Creating and Graphing Equations Using Standard Form

Key Concepts, continued• The equation of the parabola is y = x2 – 2x – 4. • Note that the y-intercept of this equation is –4 and the

constant term of the quadratic equation is –4. • The y-intercept of a quadratic is the c value of the

quadratic equation when written in standard form. • The axis of symmetry of a parabola is the line

through the vertex of a parabola about which the parabola is symmetric.

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5.3.1: Creating and Graphing Equations Using Standard Form

Key Concepts, continued

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5.3.1: Creating and Graphing Equations Using Standard Form

Key Concepts, continued• The axis of symmetry extends through the vertex of

the graph. • The vertex of a parabola is the point on a parabola

where the graph changes direction, (h, k), where h is the x-coordinate and k is the y-coordinate.

• The equation of the axis of symmetry is

• In the example above, the equation of the axis of symmetry is x = 1 because the vertical line through 1 is the line that cuts the parabola in half.

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5.3.1: Creating and Graphing Equations Using Standard Form

Key Concepts, continued• The axis of symmetry is sometimes written as

where h is the x-coordinate of the vertex. • The vertex of a quadratic lies on the axis of symmetry,

as shown in the graph on the next slide.

• The formula is not only used to find the

equation of the axis of symmetry, but also to find the x-coordinate of the vertex.

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5.3.1: Creating and Graphing Equations Using Standard Form

Key Concepts, continued

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5.3.1: Creating and Graphing Equations Using Standard Form

Key Concepts, continued• To find the y-coordinate, substitute the value of x into

the original function,

• The vertex gives information about the maximum or minimum value of a quadratic.

• The maximum is the largest y-value of a quadratic and the minimum is the smallest y-value.

• One might see these terms in real-world situations such as maximizing profit and minimizing cost, among others.

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5.3.1: Creating and Graphing Equations Using Standard Form

Key Concepts, continued• From the equation of a function in standard form, you

can determine if you have a maximum or minimum based on the sign of the coefficient of the quadratic term, a.

• If a > 0, the parabola opens up and therefore has a minimum value.

• If a < 0, the parabola opens down and therefore has a maximum value.

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5.3.1: Creating and Graphing Equations Using Standard Form

Key Concepts, continued

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5.3.1: Creating and Graphing Equations Using Standard Form

Minimum Maximum

y = 3x2 a = 3 and a > 0

y = –3x2 a = –3 and a < 0

Key Concepts, continued• To create the equation of a quadratic in standard form,

one must know some combination of the following key features: y-intercept, axis of symmetry or the vertex, and maximum or minimum.

• The key features of a quadratic are the x-intercepts, y-intercept, where the function is increasing and decreasing, where the function is positive and negative, relative minimums and maximums, symmetries, and end behavior of the function.

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5.3.1: Creating and Graphing Equations Using Standard Form

Key Concepts, continued

• To create the equation of a quadratic given the vertex

and y-intercept, first set the x-coordinate of the vertex

equal to and isolate b. • Second, substitute the expression that is equal to b,

the y-intercept, and the coordinates of the vertex into the standard form of a quadratic equation, y = ax2 + bx + c.

• Solve the equation for a and then substitute the value of a into the equation you created in the first step to determine the value of b.

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5.3.1: Creating and Graphing Equations Using Standard Form

Common Errors/Misconceptions• forgetting to write an equation for the axis of symmetry • not finding and labeling all important points • drawing the parabola as a straight line or a series of

straight lines instead of a curve

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5.3.1: Creating and Graphing Equations Using Standard Form

Guided Practice

Example 2h(x) = 2x2 – 11x + 5 is a quadratic function. Determine the direction in which the function opens, the vertex, the axis of symmetry, the x-intercept(s), and the y-intercept. Use this information to sketch the graph.

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5.3.1: Creating and Graphing Equations Using Standard Form

Guided Practice: Example 2, continued

1. Determine whether the graph opens up or down. h(x) = 2x2 – 11x + 5 is in standard form; therefore, a = 2.

Since a > 0, the parabola opens up.

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5.3.1: Creating and Graphing Equations Using Standard Form

Guided Practice: Example 2, continued

2. Find the vertex and the equation of the axis of symmetry.

h(x) = 2x2 – 11x + 5 is in standard form; therefore,

a = 2 and b = –11.

The vertex has an x-value of 2.75. 17

5.3.1: Creating and Graphing Equations Using Standard Form

Equation to determine the vertex

Substitute 2 for a and –11 for b.

x = 2.75 Simplify.

Guided Practice: Example 2, continuedSince the input value is 2.75, find the output value by evaluating the function for h(2.75).

The y-value of the vertex is –10.125.

The vertex is the point (2.75, –10.125).

Since the axis of symmetry is the vertical line through the vertex, the equation of the axis of symmetry is x = 2.75.

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5.3.1: Creating and Graphing Equations Using Standard Form

h(x) = 2x2 – 11x + 5 Original equation

h(2.75) = 2(2.75)2 – 11(2.75) + 5 Substitute 2.75 for x.

h(2.75) = –10.125 Simplify.

Guided Practice: Example 2, continued

3. Find the y-intercept. h(x) = 2x2 – 11x + 5 is in standard form, so the y-intercept is the constant c, which is 5.

The y-intercept is (0, 5).

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5.3.1: Creating and Graphing Equations Using Standard Form

Guided Practice: Example 2, continued

4. Find the x-intercepts, if any exist. The x-intercepts occur when y = 0.

Substitute 0 for the output, h(x), and solve.

This equation is factorable, but if we cannot easily identify the factors, the quadratic formula always works.

Note both methods shown on the following slides.

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5.3.1: Creating and Graphing Equations Using Standard Form

Guided Practice: Example 2, continuedSolved by factoring:

h(x) = 2x2 – 11x + 5

0 = 2x2 – 11x + 5

0 = (2x – 1)(x – 5)

0 = 2x – 1 or 0 = x – 5

x = 0.5 or x = 5

Solved using the quadratic formula:

h(x) = 2x2 – 11x + 5

0 = 2x2 – 11x + 5

a = 2, b = –11, and c = 5 21

5.3.1: Creating and Graphing Equations Using Standard Form

Guided Practice: Example 2, continued

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5.3.1: Creating and Graphing Equations Using Standard Form

Guided Practice: Example 2, continued

The x-intercepts are (0.5, 0) and (5, 0).

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5.3.1: Creating and Graphing Equations Using Standard Form

Guided Practice: Example 2, continued

5. Plot the points from steps 2–4 and their symmetric points over the axis of symmetry. Connect the points with a smooth curve.

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5.3.1: Creating and Graphing Equations Using Standard Form

Guided Practice: Example 2, continued

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5.3.1: Creating and Graphing Equations Using Standard Form

✔

Guided Practice: Example 2, continued

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5.3.1: Creating and Graphing Equations Using Standard Form

Guided Practice

Example 5Create the equation of a quadratic function given a vertex of (2, –4) and a y-intercept of 4.

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5.3.1: Creating and Graphing Equations Using Standard Form

Guided Practice: Example 5, continued

1. Write an equation for b in terms of a.

Set the x-coordinate of the vertex equal to

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5.3.1: Creating and Graphing Equations Using Standard Form

Substitute 2 for x.

4a = –b Multiply both sides by 2a.

–4a = b Multiply both sides by –1.

b = –4a

Guided Practice: Example 5, continued

2. Substitute the expression for b from step 1, the coordinates of the vertex, and the y-intercept into the standard form of a quadratic equation.

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5.3.1: Creating and Graphing Equations Using Standard Form

y = ax2 + bx + c Standard form of a quadratic equation

y = ax2 + (–4a)x + c Substitute –4a for b.

(–4) = a(2)2 + (–4a)(2) + c Substitute the vertex (2, –4) for x and y.

Guided Practice: Example 5, continued

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5.3.1: Creating and Graphing Equations Using Standard Form

(–4) = a(2)2 + (–4a)(2) + 4 Substitute the y-intercept of 4 for c.

–4 = 4a – 8a + 4 Simplify, then solve for a.

–8 = –4a

a = 2

Guided Practice: Example 5, continued

3. Substitute the value of a into the equation for b from step 1.

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5.3.1: Creating and Graphing Equations Using Standard Form

b = –4a Equation from step 1

b = –4(2) Substitute 2 for a.

b = –8 Simplify.

Guided Practice: Example 5, continued

4. Substitute a, b, and c into the standard form of a quadratic equation.

The equation of the quadratic function

with a vertex of (2, –4) and a y-intercept

of 4 is y = 2x2 – 8x + 4.32

5.3.1: Creating and Graphing Equations Using Standard Form

y = ax2 + bx + c Standard form of a quadratic equation

y = 2x2 – 8x + 4 Substitute 2 for a, –8 for b, and 4 for c.

✔

Guided Practice: Example 5, continued

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5.3.1: Creating and Graphing Equations Using Standard Form