Before you read

Answer first — see what you already know.

A ball's height model $h (t) = - 16 t^{2} + 48 t$ has vertex at $(1.5, 36)$ . What does $36$ represent in context?

A
The time the ball is in the air
B
The initial height
C
The maximum height in feet
D
The speed at launch

Check your answer

Answer:

Before you read

Answer first — see what you already know.

A profit graph (quadratic) has x-intercepts at $x = 5$ and $x = 25$ (thousands of units). When does the company break even?

A
At 5,000 and 25,000 units
B
At 15,000 units only
C
At 5 and 25 units
D
Never

Check your answer

Answer:

Theory

What Graph Features Mean in Context

On the SAT, you'll see graphs of real-world situations and must interpret their features:

Graph feature	Mathematical meaning	Real-world meaning
Y-intercept	$f(0)$	Starting/initial value
X-intercept(s)	$f(x) = 0$	When the quantity is zero
Vertex (max)	Highest $y$ -value	Maximum height/profit/amount
Vertex (min)	Lowest $y$ -value	Minimum cost/distance/time
Increasing	$f$ goes up	Quantity is growing
Decreasing	$f$ goes down	Quantity is shrinking
Positive	$f(x) > 0$	Quantity is above zero
Negative	$f(x) < 0$	Quantity is below zero

Profit function: max profit at x = 4

Theory

Reading Between the Lines

The SAT often asks questions like:

- "During what time interval was the height increasing?" $\to$ Where is the graph going up?
- "What was the maximum revenue?" $\to$ What is the y-coordinate of the vertex?
- "When did the ball hit the ground?" $\to$ Where does the graph cross the x-axis (after launch)?
- "What does the y-intercept represent?" $\to$ What's the value at $x = 0$ (often time $= 0$ )?

Example 1

A graph shows a ball's height $h$ (feet) vs time $t$ (seconds). The graph is a downward parabola with vertex at $(2, 64)$ and x-intercepts at $(0, 0)$ and $(4, 0)$ . Interpret these features.

Y-intercept $(0, 0)$ : ball starts at ground level.

Vertex $(2, 64)$ : max height is $64$ ft at $t = 2$ sec.

Second x-intercept $(4, 0)$ : ball returns to ground at $t = 4$ sec.

Increasing on $[0, 2]$ : ball is going up.

Decreasing on $[2, 4]$ : ball is coming down.

Ball reaches max height 64 ft at 2 sec, lands at 4 sec.

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