Question

in exercises 17–24, describe the transformation of ( f(x) = x^2 ) represented by ( g ). then graph each function. (see example 2.)

1. ( g(x) = -x^2 )
2. ( g(x) = (-x)^2 )
3. ( g(x) = 3x^2 )
4. ( g(x) = \frac{1}{3}x^2 )

Explanation:

Problem 17: $g(x) = -x^2$

Step1: Identify base function

Base function: $f(x) = x^2$

Step2: Compare to transformed function

$g(x) = -f(x)$: reflection over x-axis.

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Problem 18: $g(x) = (-x)^2$

Step1: Identify base function

Base function: $f(x) = x^2$

Step2: Simplify and compare

$(-x)^2 = x^2$, so $g(x)=f(x)$: no transformation (or reflection over y-axis, which maps to itself).

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Problem 19: $g(x) = 3x^2$

Step1: Identify base function

Base function: $f(x) = x^2$

Step2: Compare to transformed function

$g(x) = 3f(x)$: vertical stretch by factor 3.

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Problem 20: $g(x) = \frac{1}{3}x^2$

Step1: Identify base function

Base function: $f(x) = x^2$

Step2: Compare to transformed function

$g(x) = \frac{1}{3}f(x)$: vertical shrink by factor $\frac{1}{3}$.

Answer:

1. Transformation: Reflection of $f(x)=x^2$ across the x-axis.
2. Transformation: No net transformation (or reflection across the y-axis, which leaves $f(x)=x^2$ unchanged, since $g(x)=x^2$).
3. Transformation: Vertical stretch of $f(x)=x^2$ by a factor of 3.
4. Transformation: Vertical shrink of $f(x)=x^2$ by a factor of $\frac{1}{3}$.

(Graphing note: For each function, plot points using the transformed rule: e.g., for 17, $(x, -x^2)$; for 18, same as $f(x)$; for 19, $(x, 3x^2)$; for 20, $(x, \frac{1}{3}x^2)$ and connect with a smooth parabola.)