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Zorluk: OrtaQuadratic Functions and Graphs

The quadratic function ff is defined by f(x)=(x−4)(x−10)f(x) = (x - 4)(x - 10). In the xyxy-plane, the graph of function gg is obtained by translating the graph of ff horizontally so that the vertex of the graph of gg lies on the yy-axis. Which of the following equations defines the function gg?

  1. A
    g(x)=x2−14x+40g(x) = x^2 - 14x + 40
  2. g(x)=x2−9g(x) = x^2 - 9Cevap
  3. C
    g(x)=x2−28x+187g(x) = x^2 - 28x + 187
  4. D
    g(x)=x2+14x+40g(x) = x^2 + 14x + 40

Cevap

The correct equation is g(x)=x2−9g(x) = x^2 - 9.
The x-coordinate of the vertex of the graph of f(x)=(x−4)(x−10)f(x) = (x - 4)(x - 10) is the average of its x-intercepts: 4+102=7\frac{4 + 10}{2} = 7. For the vertex of the graph of gg to lie on the y-axis, the x-coordinate of the new vertex must be 0. This requires translating the graph of ff to the left by 7 units, which corresponds to the transformation g(x)=f(x+7)g(x) = f(x + 7). Substituting x+7x + 7 for xx in f(x)f(x) gives g(x)=((x+7)−4)((x+7)−10)=(x+3)(x−3)=x2−9g(x) = ((x + 7) - 4)((x + 7) - 10) = (x + 3)(x - 3) = x^2 - 9.

Adım Adım Çözüm

1
Identify the x-intercepts of the function f(x)=(x−4)(x−10)f(x) = (x - 4)(x - 10).
The x-intercepts are x=4x = 4 and x=10x = 10.
The x-intercepts occur where the function equals zero, which are the roots of the factors.
2
Find the x-coordinate of the vertex of the graph of ff.
The x-coordinate of the vertex is x=7x = 7.
The axis of symmetry (and thus the x-coordinate of the vertex) is the average of the x-intercepts: 4+102=7\frac{4 + 10}{2} = 7.
3
Determine the translation needed to place the vertex on the y-axis.
Translate the graph to the left by 7 units.
The y-axis corresponds to x=0x = 0. To move the vertex from x=7x = 7 to x=0x = 0, the graph must be shifted 7 units to the left.
4
Apply the horizontal translation to find the equation for g(x)g(x).
g(x)=(x+3)(x−3)=x2−9g(x) = (x + 3)(x - 3) = x^2 - 9.
A horizontal translation of 7 units to the left is represented by g(x)=f(x+7)g(x) = f(x + 7). Substituting x+7x + 7 for xx in the equation for f(x)f(x) gives ((x+7)−4)((x+7)−10)=(x+3)(x−3)=x2−9((x + 7) - 4)((x + 7) - 10) = (x + 3)(x - 3) = x^2 - 9.

Anahtar Kavram

Identifying the vertex of a quadratic function from its factored form and applying horizontal translation rules.
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Konular

Quadratic Functions and Graphs

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