Non Linear Functions Pattern - Features of Graphs

Features of Graphs

This pattern gives you a graph of a nonlinear function — usually a parabola, exponential curve, or radical function — set in a real-world context. The question asks you to interpret a specific feature: the $y$-intercept, a marked point, or the behavior of the curve. No calculation required — just read the graph and match it to the correct verbal interpretation.

The Key Features

$y$-intercept $(0, b)$: The value of the function when $x = 0$. In context, this is the initial value — what the quantity was at the start (time $= 0$, price $= 0$, etc.).

A marked point $(a, b)$: When $x = a$, the function value is $b$. In context: "At $a$ [units of $x$], the quantity is $b$ [units of $y$]."

$x$-intercept $(c, 0)$: Where the function equals zero. In context, this is when the quantity runs out, reaches zero profit, hits the ground, etc.

Vertex (for parabolas): The maximum or minimum value of the function.

Worked Examples

Example 1. The graph shows the estimated population of a bee colony over time. What is the best interpretation of the $y$-intercept?

A) The population after 8 weeks was 500
B) The initial population was 500
C) The population increased by 500 in 8 weeks
D) The population grew at a rate of 500 per week

The $y$-intercept is the point where the curve crosses the vertical axis — that's when the number of weeks is $0$ (the start). Reading the graph, the curve starts at $y = 500$.
Gotcha: Option C confuses the initial value with a change. Option D treats it as a rate. The $y$-intercept is a single starting value, not a rate or a change.
The answer is B.

Example 2. The graph shows a company's profit over 12 months. What does the point $(6, 150)$ mean?

A) At 150 months, the profit is $6,000
B) The initial profit was $6,000, growing to $150,000
C) 6 months after launch, the profit is $150,000
D) The profit grows by $150,000 every 6 months

A point $(x, y)$ on the graph means: when the $x$-variable equals $x$, the $y$-variable equals $y$. Here: at month $6$, profit $= 150$ thousand dollars.
Gotcha: Option A swaps the coordinates. Option D treats a single point as a rate. Always check which axis represents which variable.
The answer is C.

Example 3. An exponential growth graph shows the number of users of an app over time. The graph starts at $2{,}000$ and curves upward. What does the $y$-intercept represent?

The $y$-intercept is the starting value: the app had $2{,}000$ users when tracking began ($t = 0$). It is not the growth rate, and it is not the total gained over time.

Example 4. A quadratic graph shows the height of a ball over time. The vertex is at $(3, 45)$. What does this mean?

The vertex of a downward parabola is the maximum. The ball reaches its maximum height of $45$ feet at $3$ seconds after being thrown.

What to Do on Test Day

• $y$-intercept = initial value. "At time $0$, the starting amount was..." This is the single most common question type.
• A point $(a, b)$: "When $x = a$, the value of $y$ is $b$." Plug the coordinates into the context. Don't swap them.
• Don't confuse features: The vertex is the max/min. The $x$-intercept is where the value hits zero. The $y$-intercept is the start. Each has a different interpretation.
• Watch the units: If the $y$-axis is "in thousands of dollars," then $y = 150$ means $150,000, not $150.
• Rates vs. values: A single point tells you a value at a moment. It does not tell you a rate of change or a total increase. Rate requires two points.