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Zorluk: OrtaQuadratic Functions and Graphs

In the xyxy-plane, the graph of the quadratic function f(x)=−x2+6x+7f(x) = -x^2 + 6x + 7 has a vertex at (h,k)(h, k). If the graph of ff is translated 44 units to the right and 33 units up to produce the graph of the function gg, what is the maximum value of gg?

Cevap: 19

Cevap

The maximum value of the function gg is 1919.
The vertex of the original quadratic function f(x)=−x2+6x+7f(x) = -x^2 + 6x + 7 is (3,16)(3, 16). Since the coefficient of x2x^2 is negative, the graph opens downward, making 1616 the maximum value of the function. Translating the graph 33 units up shifts all yy-values up by 33, which increases the maximum value to 16+3=1916 + 3 = 19. The horizontal translation of 44 units to the right shifts the graph horizontally but does not affect the maximum output value.

Adım Adım Çözüm

1
Find the vertex of the original quadratic function f(x)=−x2+6x+7f(x) = -x^2 + 6x + 7.
The vertex of f(x)f(x) is (3,16)(3, 16).
Rewrite the function in vertex form, f(x)=−(x−3)2+16f(x) = -(x-3)^2 + 16, to identify the vertex (h,k)(h, k) as (3,16)(3, 16).
2
Determine the maximum value of the original function f(x)f(x).
The maximum value of f(x)f(x) is 1616.
Since the leading coefficient is negative, the parabola opens downward, and the maximum value occurs at the vertex's yy-coordinate.
3
Apply the vertical translation to find the maximum value of the new function g(x)g(x).
The maximum value of gg is 1919.
Translating the graph 33 units up increases all yy-values, including the maximum value, by 33, so 16+3=1916 + 3 = 19. The horizontal translation of 44 units to the right does not affect the maximum value.

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Quadratic Functions and Graphs
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