Linear Functions

A linear function changes by the same amount every time its input goes up by $1$. On the SAT you will evaluate functions written in function notation, find a slope from two points or a table, read intercepts, and interpret what the numbers in a model like $C(n) = 32 + 5n$ mean in a real situation.

• $f(x) = mx + b$: $m$ is the slope (rate of change) and $b$ is $f(0)$, the $y$-intercept.
• $f(4)$ means "put $4$ in for $x$." The question "for what $x$ is $f(x) = 4$?" is different: set the expression equal to $4$ and solve.
• Slope is $\frac{\text{change in output}}{\text{change in input}}$. Any two points on the line give the same slope.
• If $f(x + 1) = f(x) + 3$, the slope is $3$. If $f(x + 5) = f(x) + 10$, the slope is $\frac{10}{5} = 2$.
• In a model, the slope is the rate (per minute, per mile, per item), and the intercept is the starting value when the input is $0$.

$m = \frac{y_2 - y_1}{x_2 - x_1}$

• The $x$-intercept is where $f(x) = 0$: solve $mx + b = 0$ to get $x = -\frac{b}{m}$.
• A line through $(a, 0)$ and $(0, c)$ has slope $\frac{c - 0}{0 - a} = -\frac{c}{a}$.

The function $f$ is defined by $f(x) = 7 - 2x$. Find $f(-3)$, and find the value of $x$ for which $f(x) = -5$.

1. $f(-3) = 7 - 2(-3) = 7 + 6 = 13$.
2. Set $7 - 2x = -5$. Subtract $7$: $-2x = -12$. Divide: $x = 6$.

Check: $f(6) = 7 - 12 = -5$. Correct.

A climbing gym's monthly cost $C$, in dollars, is a linear function of the number of sessions $n$. A member who climbs $3$ times pays $47, and a member who climbs $8$ times pays $72. What does the gym charge for $12$ sessions, and what do the slope and intercept mean?

1. Slope: $\frac{72 - 47}{8 - 3} = \frac{25}{5} = 5$. Each session adds $5.
2. Intercept: go back $3$ sessions from $(3, 47)$: $C(0) = 47 - 3(5) = 32$. This is a $32 monthly base fee.
3. So $C(n) = 5n + 32$, and $C(12) = 60 + 32 = 92$. The cost is $92.

Check: $C(8) = 40 + 32 = 72$. Correct.

• Flipping the slope fraction. Slope is rise over run, $\frac{\Delta y}{\Delta x}$, not $\frac{\Delta x}{\Delta y}$.
• Mismatched subtraction order. If you compute $y_2 - y_1$ on top, use $x_2 - x_1$ on the bottom, in the same order.
• Confusing $f(4)$ with $f(x) = 4$. One is an input, the other is an output.
• Misreading a negative rate. In $D(t) = 900 - 25t$, the $25$ means the amount decreases by $25$ each minute. Answer choices often describe it as an increase, or as the starting amount.
• Using the intercept as the rate. The constant term is a one-time amount, not a per-unit amount.