Linear Functions Pattern - Graphical Analysis

This pattern gives you a graph of a linear function and asks you to read information from it — intercepts, specific values, slope interpretation, or the equation of a related function.

 

Finding the Y-Intercept

The y-intercept is where the line crosses the vertical axis (where $x = 0$). Read the $y$-coordinate at that point.

The graph of the function $g$ is shown, where $y = g(x)$. What is the $y$-intercept of the graph?

A) $(0, 6)$
B) $(0, 3)$
C) $(0, -6)$
D) $(0, 5)$

The line crosses the $y$-axis at $y = 6$, so the $y$-intercept is $(0, 6)$. The answer is A.

Remember: the y-intercept is always written as a point $(0, \text{something})$.

 

Finding the X-Intercept

The x-intercept is where the line crosses the horizontal axis (where $y = 0$). Read the $x$-coordinate at that point.

The graph of the linear function $h$ is shown, where $y = h(x)$. What is the $x$-intercept of the graph of $h$?

A) $(3, 0)$
B) $(0, -6)$
C) $(-2, 0)$
D) $(2, 0)$

The line crosses the $x$-axis at $x = 2$, so the $x$-intercept is $(2, 0)$. The answer is D.

Option B is the $y$-intercept, not the $x$-intercept — a common mix-up.

Key distinction: the $x$-intercept has the form $(\text{something}, 0)$ and the $y$-intercept has the form $(0, \text{something})$. Don't swap them.

 

Finding the Y-Intercept in Context

When the axes have real-world labels (like "Cost" and "Distance"), the intercept question uses those labels instead of $x$ and $y$.

The graph models the total cost $C$, in dollars, of a taxi ride for a distance of $d$ miles. What is the $C$-intercept of the graph?

A) $(0, 4)$
B) $(0, 2.5)$
C) $(-1.6, 0)$
D) $(2.5, 0)$

The "$C$-intercept" means the value of $C$ when $d = 0$ — where the line meets the vertical axis. The graph shows the line starts at $C = 4$ when $d = 0$. So the $C$-intercept is $(0, 4)$. The answer is A.

This represents the base fare before any distance is traveled. Options C and D describe horizontal-axis crossings, which would be the $d$-intercept.

 

Reading a Specific Value from the Graph

These questions ask: "What is $y$ when $x$ equals some value?" Find the input on the horizontal axis, go up (or down) to the line, then read across to the vertical axis.

A botanist tracks the growth of a plant. The graph shows the height $h$, in centimeters, of the plant $w$ weeks after germination. According to the graph, what is the height of the plant 6 weeks after germination?

A) 30
B) 45
C) 6
D) 40

Locate $w = 6$ on the horizontal axis. Move up to the line, then read across to the vertical axis: the height is 40 cm. The answer is D.

The graph shows the remaining battery percentage $y$ of a smartphone after $x$ hours of continuous use. What is the estimated battery percentage after 5 hours of use?

A) 50
B) 100
C) 80
D) 10

At $x = 5$ hours, the line is at $y = 50$. The answer is A. Option B (100) is the starting value at $x = 0$, not the value at 5 hours.

 

SPR: Reading an Initial Value

Student-produced response versions often ask for the "initial" or "starting" value, which is the y-intercept read directly from the graph.

The graph shows a model for the water depth $D$, in feet, of a reservoir $t$ days after the start of a dry season. What was the water depth, in feet, at the start of the dry season?

"At the start" means $t = 0$. Read the vertical axis where the line begins: $D = 120$. The answer is $\boldsymbol{120}$.

For any "initial value" question, just find where the line meets the vertical axis.

 

Interpreting the Slope

Medium questions often ask what the slope means in context rather than asking you to calculate it. The slope is the rate of change — the amount the output changes per one unit of input.

The graph of the linear function $f$ shows the total cost $y$, in dollars, for a gym membership over $x$ months. What is the best interpretation of the slope of the graph?

A) The one-time sign-up fee is $75.
B) The monthly membership fee is $75.
C) The one-time sign-up fee is $35.
D) The monthly membership fee is $35.

The slope is the rate of change: how much the cost increases per month. Using two points from the graph, such as $(0, 75)$ and $(10, 425)$:

$$m = \dfrac{425 - 75}{10 - 0} = \dfrac{350}{10} = 35$$

The slope is 35, meaning the cost increases by $35 each month. The answer is D.

Option A ($75) is the $y$-intercept — the sign-up fee, not the monthly rate.

Slope vs. intercept in context: the slope is always a rate (per month, per mile, per hour). The y-intercept is a starting value or fixed fee.

A car travels at a constant speed away from a city. The graph shows the car's distance from the city after $t$ hours of travel. Which is the best interpretation of the slope?

A) The total distance traveled by the car.
B) The car's initial distance from the city.
C) The speed of the car, in miles per hour.
D) The total duration of the trip.

The slope of a distance-vs-time graph is $\dfrac{\text{change in distance}}{\text{change in time}}$, which is speed. The answer is C.

Option B describes the y-intercept. Options A and D describe single values, not rates.

 

Finding f(x) from a Shifted Graph

Medium and Hard questions show the graph of $y = f(x) + k$ (or $y = f(x) - k$) and ask you to find $f(x)$. The strategy: read the graphed line's equation, then undo the shift.

The graph of $g(x) = f(x) + 9$ is shown. Which equation defines $f$?

A) $f(x) = -2x + 9$
B) $f(x) = -2x$
C) $f(x) = -\dfrac{1}{2}x - 9$
D) $f(x) = -2x - 9$

Step 1 — read the graph. The graphed line passes through $(0, 0)$ and $(-3, 6)$. Its slope is $\dfrac{6 - 0}{-3 - 0} = -2$, and its y-intercept is $0$. So the graphed function is $g(x) = -2x$.

Step 2 — undo the shift. Since $g(x) = f(x) + 9$, we get $f(x) = g(x) - 9 = -2x - 9$. The answer is D.

 

Determining Signs from a Shifted Graph (Hard)

The hardest version shows $y = f(x) + k$ (or $y = f(x) - k$) and asks which equation with positive constants $c$ and $d$ could define $f$. You don't need exact numbers — just determine whether the slope and intercept of $f$ are positive or negative.

The graph of $y = f(x) - 20$ is shown. If $c$ and $d$ are positive constants, which equation could define $f$?

A) $f(x) = -d - cx$
B) $f(x) = d - cx$
C) $f(x) = d + cx$
D) $f(x) = -d + cx$

Step 1 — read the graph's slope and intercept signs. The graphed line has a negative slope (going down left to right) and a negative y-intercept (crosses the $y$-axis below zero). Call the graphed function $g(x) = mx + b$ where $m < 0$ and $b < 0$.

Step 2 — undo the shift. Since $g(x) = f(x) - 20$, we get $f(x) = g(x) + 20 = mx + (b + 20)$.

The slope stays the same (negative), so the $x$-coefficient must be $-c$ (negative, since $c > 0$). Adding 20 to a negative intercept could make it positive (if $|b| < 20$) — so the constant term is $+d$. This gives $f(x) = d - cx$. The answer is B.

Strategy for sign-determination questions: (1) note the slope sign and y-intercept sign from the graph, (2) apply the vertical shift to adjust the intercept, (3) match the resulting signs to the answer choices. Since $c$ and $d$ are positive, $-cx$ means negative slope and $+d$ means positive intercept.

 

Quick Reference

What's asked	Where to look
$y$-intercept	Where the line crosses the vertical axis
$x$-intercept	Where the line crosses the horizontal axis
Value at $x = a$	Go to $a$ on horizontal axis, read up to the line
Slope meaning	Rate of change (per unit of input)
$f(x)$ from $g(x) = f(x) + k$	Find $g(x)$ from graph, then $f(x) = g(x) - k$