2.11 Warm UpGraph the functions & compare

to the parent function, y = x². Find the vertex, axis of symmetry, domain & range.

1. y = x² - 2

2. y = 2x²

3. y = 1/3x² + 3

2.11 Graph y = ax2 + bx + c

Vocabulary

If a>0, then the y coordinate of the vertex is the minimum value.

If a<0, the y coordinate of the vertex is the maximum value.

The x coordinate of the vertex is x =

b 2a

–

EXAMPLE 1 Find the axis of symmetry and the vertex

122(– 2)

x = – - b 2a

= = 3 Substitute – 2 for a and 12 for b. Then simplify.

For the function y = –2x2 + 12x – 7a.

b. The x-coordinate of the vertex is , or 3. b 2a

–

y = –2(3)2 + 12(3) – 7 = 11 Substitute 3 for x. Then simplify.

ANSWER The vertex is (3, 11).

a = 2 and b = 12.

EXAMPLE 2 Graph y = ax2 + bx + c

Graph y = 3x2 – 6x + 2.

Determine whether the parabola opens up or down. Because a > 0, the parabola opens up.

STEP 1

STEP 2Find and plot the vertex.

EXAMPLE 2

To find the y-coordinate, substitute 1 for x in the function and simplify.

y = 3(1)2 – 6(1) + 2 = – 1

So, the vertex is (1, – 1).

STEP 3Plot two points. Choose two x-values one on each side of the vertex. Then find the corresponding y-values.

Graph y = ax2 + b x + c

The x-coordinate of the vertex is b2a

, or 1.–

EXAMPLE 2 Standardized Test Practice

x 0 2

y 2 2

STEP 4Draw a parabola through the plotted points.

STEP 5

Find the axis of symmetry.

GUIDED PRACTICE for Examples 1 and 2

1. Find the axis of symmetry and vertex of the graph of the function y = x2 – 2x – 3.

ANSWER x = 1, (1, –4).

2. Graph the function y = 3x2 + 12x – 1. Label the vertex and axis of symmetry.

ANSWER

EXAMPLE 3 Find the minimum or maximum value

Tell whether the function f(x) = – 3x2 – 12x + 10 has aminimum value or a maximum value. Then find theminimum or maximum value.

SOLUTION

Because a = – 3 and – 3 < 0, the parabola opens down andthe function has a maximum value. To find the maximumvalue, find the vertex.

x = – = – = – 2 b2a

– 122(– 3)

f(–2) = – 3(–2)2 – 12(–2) + 10 = 22 Substitute –2 for x. Thensimplify.

The x-coordinate is – b2a

The maximum value of the function is f(– 2) = 22.

Find the minimum value of a functionEXAMPLE 4

The suspension cables between the two towers of the Mackinac Bridge in Michigan form a parabola that can be modeled by the graph of y = 0.000097x2 – 0.37x + 549 where x and y are measured in feet. What is the height of the cable above the water at its lowest point?

SUSPENSION BRIDGES

Find the minimum value of a functionEXAMPLE 4

SOLUTION

The lowest point of the cable is at the vertex of theparabola. Find the x-coordinate of the vertex. Use a = 0.000097 and b = – 0.37.

x = – = – ≈ 1910 b2a

– 0.372(0.000097)

Use a calculator.

Substitute 1910 for x in the equation to find they-coordinate of the vertex.

y ≈ 0.000097(1910)2 – 0.37(1910) + 549 ≈ 196

The cable is about 196 feet above the water at its lowest point.

GUIDED PRACTICE for Examples 3 and 4

3. Tell whether the function f(x) = 6x2 + 18x + 13 has aminimum value or a maximum value. Then find theminimum or maximum value.

1 2

Minimum value;

ANSWER

4. The cables between the two towers of the Takoma Narrows Bridge form a parabola that can be modeled by the graph of the equation y = 0.00014x2 – 0.4x + 507 where x and y are measured in feet. What is the height of the cable above the water at its lowest point? Round your answer to the nearest foot.

ANSWER 221 feet