Question

(a) complete the statements below about the graphs of $y = \frac{1}{5}x$ and $y = x$.

compared to the graph of $y = x$, the graph of $y = \frac{1}{5}x$ is select

compared to the graph of $y = x$, the graph of $y = \frac{1}{5}x$ intersects the $y$-axis at select
(b) complete the statements below about the graphs of $y = x + 1$ and $y = x$.

compared to the graph of $y = x$, the graph of $y = x + 1$ is select

compared to the graph of $y = x$, the graph of $y = x + 1$ intersects the $y$-axis at select

Explanation:

(a) Step1: Analyze slope steepness

For linear functions $y=mx+b$, $m$ is the slope. $y=x$ has $m=1$, $y=\frac{1}{5}x$ has $m=\frac{1}{5}$. Since $0<\frac{1}{5}<1$, the line is less steep (vertically compressed).

(a) Step2: Analyze y-intercept

Both functions have $b=0$, so they cross the y-axis at $(0,0)$.

(b) Step1: Analyze vertical shift

$y=x+1$ has the same slope $m=1$ as $y=x$, but $b=1$. A positive $b$ value shifts the line upward by $b$ units.

(b) Step2: Analyze y-intercept

$y=x$ has y-intercept $(0,0)$, $y=x+1$ has y-intercept $(0,1)$, so it intersects the y-axis at a higher point.

Answer:

(a)

1. Compared to the graph of $y=x$, the graph of $y=\frac{1}{5}x$ is less steep (a vertical compression)
2. Compared to the graph of $y=x$, the graph of $y=\frac{1}{5}x$ intersects the y-axis at the same point $(0,0)$

(b)

1. Compared to the graph of $y=x$, the graph of $y=x+1$ is shifted up 1 unit
2. Compared to the graph of $y=x$, the graph of $y=x+1$ intersects the y-axis at $(0,1)$ (a higher point)