The image of the function f(x) after vertical shrink by a factor of 1/2
and a reflection in the y-axis, followed by a translation 1 unit down
is g(x) = [tex]\frac{1}{2}[/tex] x² - 1
Step-by-step explanation:
Lets revise:
1. The vertical shrink
A vertical shrinking is the squeezing of the graph toward the x-axis.
if 0 < k < 1 (a fraction), the graph of y = k•f(x) is the graph of f(x) vertically
shrunk by multiplying each of its y-coordinates by k
2. The reflection
If the function f(x) reflected across the y-axis, then its image is g(x) = f(-x)
3. Vertical translation
If the function f(x) translated vertically up by m units, then its image
is g(x) = f(x) + m
If the function f(x) translated vertically down by m units, then its image
is g(x) = f(x) - m
Now let us solve the problem
∵ f(x) = x²
∵ f(x) shrunk by a factor of [tex]\frac{1}{2}[/tex]
∴ The image of f(x) = [tex]\frac{1}{2}[/tex] x²
∵ The image of f(x) reflected across y-axis
∴ The sign of x will change
∴ The new image of f(x) = [tex]\frac{1}{2}[/tex] (-x)²
∵ The new image of f(x) translated 1 unit down
∴ We will subtract the new image of f(x) by 1
∴ The last image of f(x) is g(x) = [tex]\frac{1}{2}[/tex] (-x)² - 1
V.I.Note:
(-x)² = x² because even exponents reject the negative sign
The image of the function f(x) after vertical shrink by a factor of 1/2
and a reflection in the y-axis, followed by a translation 1 unit down
is g(x) = [tex]\frac{1}{2}[/tex] x² - 1
The attached graph for more understand
Learn more:
you can learn more about transformation in brainly.com/question/2415963
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