introduction the equation of a quadratic function can be written in several different forms. we have...

IntroductionThe equation of a quadratic function can be written in several different forms. We have practiced using the standard form of a quadratic function to identify key information and graph the function. In this lesson, we will explore writing quadratic functions in intercept form or factored form, graphing a quadratic function from intercept form, and creating the equation of a quadratic function given its graph. An equation written in intercept form allows you to easily identify the x-intercepts of a quadratic function.

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5.3.2: Creating and Graphing Equations Using the x-intercepts

Key Concepts• A quadratic function in standard form can be created

from the intercept form.

• The intercept form of a quadratic function is f(x) = a(x – p)(x – q), where p and q are the zeros of the function.

• To write a quadratic in intercept form, distribute the terms of the first set of parentheses over the terms of the second set of parentheses and simplify.

• For example, f(x) = 3(x – 4)(x – 1) becomes f(x) = 3x2 – 15x + 12.

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Key Concepts, continued• The y-intercept of a quadratic function written in

intercept form can be found by substituting 0 for x.

• The x-intercept(s) of a quadratic function written in intercept form can be found by substituting 0 for y.

• The x-intercepts, also known as the zeros, roots, or solutions of the quadratic, are often identified as p and q.

• The axis of symmetry is located halfway between the zeros. To determine its equation, use the formula

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Key Concepts, continued• Recall that the axis of symmetry extends through the

vertex of a quadratic function.

• As such, the x-coordinate of the vertex is the value of the equation of the axis of symmetry.

• If the vertex is the point (h, k), then the equation of the axis of symmetry is x = h.

• The y-value can be found by substituting the x-value into the equation.

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Common Errors/Misconceptions• multiplying out the factored form when it is more

advantageous not to • drawing a parabola as a straight line or a series of lines

instead of a curve

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Guided Practice

Example 2Identify the x-intercepts, the axis of symmetry, and the vertex of the quadratic f(x) = (x + 5)(x + 2). Use this information to graph the function.

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Guided Practice: Example 2, continued

1. Identify the x-intercepts and plot the points. The x-intercepts of the quadratic are the zeros or solutions of the quadratic.

Start by setting the equation equal to 0 and solving to determine the zeros.

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f(x) = (x + 5)(x + 2) Original equation

0 = (x + 5)(x + 2) Set the equation equal to 0.

0 = x + 5 or 0 = x + 2 Set each factor equal to 0.

–5 = x or –2 = x Solve each equation for x.


2. Determine the axis of symmetry to find the vertex. The axis of symmetry is the line that divides the parabola in half.

To determine its equation, find the midpoint of the two zeros and draw the vertical line through that point.

Insert the values of x into the formula to find the midpoint.

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Guided Practice: Example 2, continuedLet p = –5 and q = –2.

The equation of the axis of symmetry is x = –3.5.

The axis of symmetry extends through the vertex of the quadratic. This means that the vertex (h, k) of the graph has an x-value of –3.5.

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Guided Practice: Example 2, continuedSubstitute –3.5 for x in the original equation to determine the y-value of the vertex.

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f(x) = (x + 5)(x + 2) Original equation

f(–3.5) = (–3.5 + 5)(–3.5 + 2) Substitute –3.5 for x.

f(–3.5) = (1.5)(–1.5) Simplify.

f(–3.5) = –2.25

Guided Practice: Example 2, continuedThe y-value of the vertex is –2.25.

The vertex is the point (–3.5, –2.25).

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3. Graph the zeros, the vertex, and the axis of symmetry of the given equation.

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4. Sketch the function based on the zeros and the vertex.

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✔


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Guided Practice

Example 3Use the intercepts and a point on the graph at right to write the equation of the function in standard form.

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1. Find the x-intercepts of the graph and write them as zeros of the equation. Identify the two points of intersection and write them as zeros to prepare to write the factors of the equation.

The x-intercepts are (–2, 0) and (3, 0).

The zeros are x = –2 and x = 3.

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2. Write the zeros as expressions equal to 0 to determine the factors of the quadratic.

x = –2 x = 3

x + 2 = 0 x – 3 = 0

The factors of the equation are (x + 2) and (x – 3), so the equation is f(x) = a(x + 2)(x – 3).

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3. Use the point (4, –3) to find a, the coefficient of the x2 term. The equation is f(x) = a(x + 2)(x – 3) is written in intercept form, or f(x) = a(x – p)(x – q).

Substitute (4, –3) into the equation to find a.

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The coefficient of x2 is

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f(x) = a(x + 2)(x – 3) Equation

–3 = a(4 + 2)(4 – 3) Substitute (4, –3) for x and f(x).

–3 = a(6)(1) Simplify.

–3 = 6a


4. Write the equation in standard form. Insert the value found for a into the equation from step 1 and solve.

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f(x) = a(x + 2)(x – 3) Equation

Substitute for a.


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Simplify by multiplying the factors.

Distribute over the

parentheses.

The coefficient of x2 is ✔


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introduction the equation of a quadratic function can be written in several different forms. we have...

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