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Zorluk: OrtaQuadratic Functions and Graphs

The graph of the quadratic function ff in the xyxy-plane is defined by f(x)=x2−6x+5f(x) = x^2 - 6x + 5. If the graph of ff is translated 33 units to the left and 22 units down, which of the following equations defines the resulting function gg?

  1. A
    g(x)=x2−12x+30g(x) = x^2 - 12x + 30
  2. g(x)=x2−6g(x) = x^2 - 6Cevap
  3. C
    g(x)=x2−2g(x) = x^2 - 2
  4. D
    g(x)=x2−12x+34g(x) = x^2 - 12x + 34

Cevap

The equation that defines the function gg is g(x)=x2−6g(x) = x^2 - 6.
To find the function g(x)g(x), we apply the transformations to f(x)=x2−6x+5f(x) = x^2 - 6x + 5. A translation of 33 units to the left replaces xx with x+3x + 3, which gives f(x+3)=(x+3)2−6(x+3)+5=x2+6x+9−6x−18+5=x2−4f(x + 3) = (x + 3)^2 - 6(x + 3) + 5 = x^2 + 6x + 9 - 6x - 18 + 5 = x^2 - 4. Next, translating the graph 22 units down corresponds to subtracting 22 from the function, resulting in g(x)=(x2−4)−2=x2−6g(x) = (x^2 - 4) - 2 = x^2 - 6.

Adım Adım Çözüm

1
Rewrite the function f(x)f(x) in vertex form by completing the square.
f(x)=(x−3)2−4f(x) = (x - 3)^2 - 4
Converting to vertex form a(x−h)2+ka(x - h)^2 + k makes it straightforward to apply horizontal and vertical shifts based on the vertex (h,k)(h, k).
2
Apply the horizontal translation of 33 units to the left by replacing xx with x+3x + 3 in the function.
f(x+3)=((x+3)−3)2−4=x2−4f(x + 3) = ((x + 3) - 3)^2 - 4 = x^2 - 4
A horizontal shift of cc units to the left is represented by replacing xx with x+cx + c.
3
Apply the vertical translation of 22 units down by subtracting 22 from the expression obtained in the previous step.
g(x)=(x2−4)−2=x2−6g(x) = (x^2 - 4) - 2 = x^2 - 6
A vertical shift of dd units downward is represented by subtracting dd from the function's output.

Anahtar Kavram

Translations of quadratic functions in the coordinate plane.
Tahmini Süre:1m 30s
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Konular

Quadratic Functions and Graphs

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