2.11 warm up graph the functions & compare to the parent function, y = x². find the vertex,...
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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
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