Graphing Linear Functions

−2

Find b: $-9 = -3(1) + b$, so $b = -6$ and $f(x) = -3x - 6$. Set $f(x) = 0$: $-3x - 6 = 0$, so $x = -2$.

2: The sign is wrong. Solve $-3x - 6 = 0$ carefully: $x = -2$.

4: Check how you used the slope. Going from $(1, -9)$ up to $y = 0$ with slope −3 moves left, not right.

−6: −6 is the y-intercept. The zero is the x-value where $f(x) = 0$.

The x-intercept of the graph of $k$ is −7.

The point $(-7, 0)$ has a y-value of 0, so the graph crosses the x-axis at −7. That is the x-intercept.

The slope of the graph of $k$ is $-\frac{4}{3}$.: A sign error was made. The slope is $\frac{8 - 0}{1 - (-7)} = \frac{8}{8} = 1$.

The graph of $k$ passes through the point $(-1, -8)$.: This flips the signs of $(1, 8)$. That point isn't on the line.

The zero of $k$ is 7.: The zero is where $y = 0$, at $x = -7$, not 7.

Graph C

Divide by −5: $y = \frac{6}{5}x - 3$. Graph C has slope $\frac{6}{5}$ and y-intercept −3.

Graph A: This line has a negative slope, but the simplified equation has slope $\frac{6}{5}$.

Graph B: This line has a negative slope instead of $\frac{6}{5}$.

Graph D: This line has y-intercept 3 rather than −3.

Graph H

The slope is −4 and the line passes through $(-3,-2)$. Graph H has that negative slope and contains $(-3,-2)$.

Graph F: This line rises from left to right, but the slope is −4.

Graph G: This line has a positive slope; the given slope is −4.

Graph J: This negative-slope line does not pass through $(-3,-2)$.

Graph G

The line must pass through $(-2,0)$ and $(0,6)$. Its slope is $\frac{6-0}{0-(-2)}=3$, matching Graph G.

Graph F: This line has a negative slope and does not pass through $(-2,0)$.

Graph H: This line has a positive slope but the wrong intercepts.

Graph J: This line has a negative slope; the required slope is 3.

The slope is $\frac{5}{7}$, and the y-intercept is 1.

Slope: $\frac{6-(-4)}{7-(-7)}=\frac{10}{14}=\frac{5}{7}$. Using $(7,6)$ gives $6=\frac{5}{7}(7)+b$, so $b=1$.

The slope is $\frac{5}{7}$, and the y-intercept is −1.: The slope is correct, but the graph crosses the y-axis at 1.

The slope is $\frac{7}{5}$, and the y-intercept is −1.: The slope uses rise over run, not run over rise, and the y-intercept is 1.

The slope is $\frac{7}{5}$, and the y-intercept is 1.: The y-intercept is correct, but the slope is $\frac{10}{14}=\frac{5}{7}$.

Graph D

The graph must contain $(-1,0)$ and $(0,5)$. The slope is $\frac{5-0}{0-(-1)}=5$, matching Graph D.

Graph A: This line has a small positive slope and does not pass through $(0,5)$.

Graph B: This line has a negative slope; the required slope is 5.

Graph C: This line has a negative slope instead of 5.

The x-intercept is 60, which represents the number of monthly payments needed to repay the loan.

The x-intercept is where the remaining balance is 0. The line reaches $y=0$ at $x=60$, so 60 monthly payments repay the loan.

The x-intercept is 60, which represents the initial balance in dollars of the loan.: 60 is a number of payments, not a dollar balance; the initial balance is the y-intercept.

The x-intercept is 27,000, which represents the initial balance in dollars of the loan.: 27,000 is the initial dollar balance and y-intercept, not the x-intercept.

The x-intercept is 27,000, which represents the number of monthly payments needed to repay the loan.: The x-axis measures payments and reaches zero balance at 60, not 27,000.

A line through $(0, 2)$ and $(3, 0)$

The $y$-intercept is 2, so graph $(0, 2)$. The slope is $-\frac{2}{3}$, so moving 3 units right and 2 units down gives $(3, 0)$.

A line through $(0, 2)$ and $(3, 0)$

Setting x = 0 gives y = 2, and setting y = 0 gives x = 3. Plot $(0, 2)$ and $(3, 0)$ and connect them. Equivalently, the line is $y = -\frac{2}{3}x + 2$.

–5

The zero is the $x$-value where $y = 0$. The line crosses the $x$-axis at $(-5, 0)$, so the zero is −5.

2: This is the slope of the line, not the zero.

10: This is the $y$-intercept, where the line crosses the $y$-axis.

–2: This is the opposite of the slope, not the zero.

Graph D

The line crosses the $y$-axis at $(0, 4)$. A slope of $\frac{2}{3}$ means moving 3 units right and 2 units up, so $(3, 6)$ is also on the line. Graph D shows this line.

Graph A: This line passes through $(4, 0)$, treating the $y$-intercept as an $x$-intercept, and it slopes down.

Graph B: This line passes through $(0, 4)$, but it moves 2 units down instead of up for every 3 units right.

Graph C: This line treats the $y$-intercept as the $x$-intercept $(4, 0)$.

1

The line crosses the $y$-axis at $(0, 1)$, so the $y$-intercept is 1.

3: This is the $x$-intercept, where the line crosses the $x$-axis.

–1: This uses the wrong sign for the point $(0, 1)$.

–3: This treats the $x$-intercept as the $y$-intercept and uses the wrong sign.