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2.1 a and b. Finding the vertex and y-intercept from standard form Graphing in standard form. These properties can be generalized to help you graph quadratic functions. Helpful Hint. When a is positive, the parabola is happy (U). When the a negative, the parabola is sad ( ). U.

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2 1 a and b

2.1 a and b

Finding the vertex and y-intercept from standard form

Graphing in standard form

slide3

Helpful Hint

When a is positive, the parabola is happy (U). When the a negative, the parabola is sad ( ).

U

The Axis of Symmetry is the same as the x-coordinate of the vertex.

slide4

The axis of symmetry is given by .

Example 2A: Graphing Quadratic Functions in Standard Form

Consider the function f(x) = 2x2 – 4x + 5.

a. Determine whether the graph opens upward or downward.

Because a is positive, the parabola opens upward.

b. Find the axis of symmetry.

Substitute –4 for b and 2 for a.

The axis of symmetry is the line x = 1.

slide5

Example 2A: Graphing Quadratic Functions in Standard Form

Consider the function f(x) = 2x2 – 4x + 5.

c. Find the vertex.

The vertex lies on the axis of symmetry, so the x-coordinate is 1. The y-coordinate is the value of the function at this x-value, or f(1).

f(1) = 2(1)2 – 4(1) + 5 = 3

The vertex is (1, 3).

d. Find the y-intercept.

Because c = 5, the intercept is 5.

slide6

Example 2A: Graphing Quadratic Functions in Standard Form

Consider the function f(x) = 2x2 – 4x + 5.

e. Graph the function.

Graph by making a table of values with the x-coordinate of the vertex in the center.

slide7

The axis of symmetry is given by .

Example 2B: Graphing Quadratic Functions in Standard Form

Consider the function f(x) = –x2 – 2x + 3.

a. Determine whether the graph opens upward or downward.

Because a is negative, the parabola opens downward.

b. Find the axis of symmetry.

Substitute –2 for b and –1 for a.

The axis of symmetry is the line x = –1.

slide8

Example 2B: Graphing Quadratic Functions in Standard Form

Consider the function f(x) = –x2 – 2x + 3.

c. Find the vertex.

The vertex lies on the axis of symmetry, so the x-coordinate is –1. The y-coordinate is the value of the function at this x-value, or f(–1).

f(–1) = –(–1)2 – 2(–1) + 3 = 4

The vertex is (–1, 4).

d. Find the y-intercept.

Because c = 3, the y-intercept is 3.

slide9

Example 2B: Graphing Quadratic Functions in Standard Form

Consider the function f(x) = –x2 – 2x + 3.

e. Graph the function.

Graph by making a table of values with the x-coordinate of the vertex in the center.

slide10

b. The axis of symmetry is given by .

Check It Out! Example 2a

For the function, (a) determine whether the graph opens upward or downward, (b) find the axis of symmetry, (c) find the vertex, (d) find the y-intercept, and (e) graph the function.

f(x)= –2x2 – 4x

a. Because a is negative, the parabola opens downward.

Substitute –4 for b and –2 for a.

The axis of symmetry is the line x = –1.

slide11

Check It Out! Example 2a

f(x)= –2x2 – 4x

c. The vertex lies on the axis of symmetry, so the x-coordinate is –1. The y-coordinate is the value of the function at this x-value, or f(–1).

f(–1) = –2(–1)2 – 4(–1) = 2

The vertex is (–1, 2).

d. Because c is 0, the y-intercept is 0.

slide12

Check It Out! Example 2a

f(x)= –2x2 – 4x

e. Graph the function.

Graph by making a table of values with the x-coordinate of the vertex in the center.

slide13

b. The axis of symmetry is given by .

The axis of symmetry is the line .

Check It Out! Example 2b

For the function, (a) determine whether the graph opens upward or downward, (b) find the axis of symmetry, (c) find the vertex, (d) find the y-intercept, and (e) graph the function.

g(x)= x2 + 3x – 1.

a. Because a is positive, the parabola opens upward.

Substitute 3 for b and 1 for a.

slide14

c. The vertex lies on the axis of symmetry, so the x-coordinate is . The y-coordinate is the value of the function at this x-value, or f().

f( ) = ( )2 + 3( ) – 1 =

The vertex is ( , ).

Check It Out! Example 2b

g(x)= x2 + 3x – 1

d. Because c = –1, the intercept is –1.

slide15

Check It Out! Example2

g(x)= x2 + 3x – 1

e. Graph the function.

Graph by making a table

of values with the x-coordinate

of the vertex in the center.

slide16

Lesson Quiz: Part I

Consider the function f(x)= 2x2 + 6x – 7.

1. Determine whether the graph opens upward or downward.

2. Find the axis of symmetry.

3. Find the vertex.

4. Identify the maximum or minimum value of the function.

5. Find the y-intercept.

upward

x = –1.5

(–1.5, –11.5)

min.: –11.5

–7

slide17

Lesson Quiz: Part II

Consider the function f(x)= 2x2 + 6x – 7.

6. Graph the function.

By making a table

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