Question

1. determine the transformations in each function as it relates to the parent function f(x)=x². match each graph to the correct function.

a. g(x)=(x - 5)²-3
b. k(x)=(x + 5)²-3
c. h(x)=(x + 3)²-5
d. j(x)=(x - 3)²-5

1. name and describe two types of transformations.
2. identify the reflection in each function as it relates to the graph of its parent function.

a. g(x)=-|2x| a reflection in the x - axis b reflection in the y - axis
b. h(x)=-2x² a reflection in the x - axis b reflection in the y - axis
c. k(x)=-3|-x + 4| a reflection in the x - axis b reflection in the y - axis

1. describe the transformations in g(x)=2|x - 1|+5 as it relates to the graph of the parent function.

Explanation:

Step1: Recall transformation rules

For a function $y = a(x - h)^2+k$, $h$ represents horizontal shift and $k$ represents vertical shift. For $y = -f(x)$ it's a reflection over the $x -$axis and for $y = f(-x)$ it's a reflection over the $y -$axis.

Step2: Analyze $g(x)=(x - 5)^2-3$

The parent function $y=x^2$ is shifted 5 units to the right (because of $x-5$) and 3 units down (because of - 3). The vertex of $y = x^2$ is $(0,0)$ and the vertex of $g(x)$ is $(5,-3)$. This matches graph D.

Step3: Analyze $k(x)=(x + 5)^2-3$

The parent function $y=x^2$ is shifted 5 units to the left (because of $x + 5$) and 3 units down (because of - 3). The vertex of $k(x)$ is $(-5,-3)$. This matches graph C.

Step4: Analyze $h(x)=(x + 3)^2-5$

The parent function $y=x^2$ is shifted 3 units to the left (because of $x + 3$) and 5 units down (because of - 5). The vertex of $h(x)$ is $(-3,-5)$. This matches graph A.

Step5: Analyze $j(x)=(x - 3)^2-5$

The parent function $y=x^2$ is shifted 3 units to the right (because of $x - 3$) and 5 units down (because of - 5). The vertex of $j(x)$ is $(3,-5)$. This matches graph B.

Step6: Answer 18

Translation: A translation moves the graph of a function horizontally, vertically or both. For example, for $y=f(x - h)+k$, $h$ horizontal shift and $k$ vertical shift. Reflection: A reflection flips the graph of a function over a line. For example, $y=-f(x)$ is a reflection over the $x -$axis.

Step7: Answer 19a

For $g(x)=-|2x|$, compared to the parent function $y = |2x|$, the negative sign in front of the absolute - value function means a reflection in the $x -$axis. So the answer is A.

Step8: Answer 19b

For $h(x)=-2x^2$, compared to the parent function $y = 2x^2$, the negative sign in front of the quadratic function means a reflection in the $x -$axis. So the answer is A.

Step9: Answer 19c

For $k(x)=-3|-x + 4|$, first, we can rewrite it as $k(x)=-3|-(x - 4)|=-3|x - 4|$. The negative sign in front of the absolute - value function means a reflection in the $x -$axis. So the answer is A.

Step10: Answer 20

For the function $g(x)=2|x - 1|+5$ compared to the parent function $y = |x|$: The factor of 2 in front of the absolute - value function vertically stretches the graph by a factor of 2. The $x-1$ inside the absolute - value function shifts the graph 1 unit to the right. The + 5 outside the absolute - value function shifts the graph 5 units up.

Answer:

17.
a. D
b. C
c. A
d. B

1. Translation: moves graph horizontally/vertically. Reflection: flips graph over a line.

19.
a. A. Reflection in the x - axis
b. A. Reflection in the x - axis
c. A. Reflection in the x - axis

1. Vertically stretched by a factor of 2, shifted 1 unit right and 5 units up.