Question

evaluate independent practice
learning goal from lesson 5.1
i can determine the effect on the graph of replacing f(x) by f(x)+k, kf(x), f(kx), and f(x + k) for specific values of k (both positive and negative). i can determine the translation value k, given a graph for slides, shifts, and stretches. i can explain the translation effects on the graph of a function using technology.
how i did (circle one)
i got it! im still learning it.
lesson 5.1 checkpoint
once you have completed the above problems and checked your solutions, complete the lesson checkpoint below.
complete the lesson reflection above by circling your current understanding of the learning goal.

1. how is the graph of g(x)=x³ + 1 related to the graph of f(x)=x³?

□ a. the graph of g(x) is a translation of the graph of f(x) down 1 unit.
□ b. the graph of g(x) is a translation of the graph of f(x) up 1 unit.
□ c. the graph of g(x) is a translation of the graph of f(x) right 1 unit.

1. how is the graph of g(x)=-(x - 3)³ related to the graph of f(x)=x³?

□ a. the graph of g(x) is a translation of the graph of f(x) right 3 units and a reflection across the y - axis.
□ b. the graph of g(x) is a translation of the graph of f(x) down 3 units and a reflection across the x - axis.
□ c. the graph of g(x) is a translation of the graph of f(x) right 3 units and a reflection across the x - axis.

1. the graph of f(x)=x³ with three reference points is shown. which of the following represents points from f(x) to a point on g(x)=2(x - 4)³ - 3? select three that apply.

□ a. (5,3)
□ b. (5, - 1)
□ c. (3, - 5)
□ d. (4, - 3)
□ e. (2, - 5)

Explanation:

Step1: Analyze transformation for question 1

For $g(x)=x^{3}+1$ compared to $f(x)=x^{3}$, the general rule for $y = f(x)+k$ is a vertical - shift. When $k = 1$, it's a shift up 1 unit.

Step2: Analyze transformation for question 2

For $g(x)=-(x - 3)^{3}$ compared to $f(x)=x^{3}$, the $x-3$ inside the function represents a right - shift of 3 units and the negative sign in front represents a reflection across the $x$-axis.

Step3: Analyze transformation for question 3

For $g(x)=2(x - 4)^{3}-3$ compared to $f(x)=x^{3}$, the transformation from $f(x)$ to $g(x)$ has a horizontal shift of 4 units to the right, a vertical shift of 3 units down and a vertical stretch by a factor of 2. For the point $(x,y)$ on $f(x)$ and $(x',y')$ on $g(x)$, $x'=x + 4$ and $y'=2y-3$. When $x = 4$ (a special point for the horizontal shift), $g(4)=2(4 - 4)^{3}-3=-3$, so $(4,-3)$ is a point on $g(x)$. When $x = 5$, $g(5)=2(5 - 4)^{3}-3=2\times1 - 3=-1$, so $(5,-1)$ is a point on $g(x)$. When $x = 2$, $g(2)=2(2 - 4)^{3}-3=2\times(-8)-3=-16 - 3=-19
eq - 5$. When $x = 3$, $g(3)=2(3 - 4)^{3}-3=2\times(-1)-3=-5$, so $(3,-5)$ is a point on $g(x)$.

Answer:

1. B. The graph of $g(x)$ is a translation of the graph of $f(x)$ up 1 unit.
2. C. The graph of $g(x)$ is a translation of the graph of $f(x)$ right 3 units and a reflection across the $x$-axis.
3. B. $(5,-1)$, C. $(3,-5)$, D. $(4,-3)$