Task Sets: Interpret key features of graphs - Common Core Math - HS Functions

Work through problems with immediate feedback

Recall / Warm-Up

What is the $y$ -intercept of the graph of $f(x) = 3x - 12$ ?

$(4, 0)$

$(0, -12)$

$(0, 3)$

$(-12, 0)$

The graph of a function passes through the points $(-3, 0)$ , $(0, 5)$ , and $(2, 0)$ . Which are the $x$ -intercepts?

$(0, 5)$ only

$(-3, 0)$ and $(2, 0)$

$(0, 5)$ , $(-3, 0)$ , and $(2, 0)$

The $y$ -coordinate 5 is the $x$ -intercept.

The function $f$ has these values: $f(1) = 5$ , $f(2) = 8$ , $f(3) = 11$ . What can you say about $f$ on the interval $[1, 3]$ ?

The function is positive on $[1, 3]$ only if the values are getting larger.

The function is increasing on $[1, 3]$ because the output grows as the input grows.

The function is positive on $[1, 3]$ because all values are greater than zero.

The function is neither increasing nor positive because we don't have a formula.

Fluency Practice

A company's monthly profit (in dollars) is $P(x) = 4x - 200$ , where $x$ is the number of items sold. Find the $x$ -intercept and enter the value of $x$ at that point.

A ball is kicked upward. Its height in meters is $h(t) = -5t^2 + 20t$ , where $t$ is time in seconds. The $y$ -intercept tells you the height when the ball is kicked. What is the $y$ -intercept (in meters)?

The graph below shows a function $f$ . On the interval $[0, 3]$ , the graph goes from $f(0) = 2$ up to $f(3) = 8$ . On the interval $[3, 6]$ , the graph goes from $f(3) = 8$ down to $f(6) = 1$ . On which interval is $f$ decreasing?

$[0, 3]$ because $f(x)$ is positive there.

$[3, 6]$ because $f(x)$ gets smaller as $x$ increases.

$[0, 3]$ because $f(x)$ goes up to 8, which is a large number.

The function is not decreasing anywhere because all values are positive.

A function $g$ has $g(x) > 0$ for $1 < x < 5$ and $g(x) < 0$ for $x < 1$ and $x > 5$ . What are the $x$ -intercepts of $g$ ?

$x = 1$ and $x = 5$

$x = 0$ and $x = 6$

There are no $x$ -intercepts because $g$ changes sign.

$x = 3$ only (the midpoint between the sign changes)

A graph shows two peaks: one at $(2, 15)$ and one at $(6, 20)$ . A student says the relative maximum is at $(6, 20)$ because it is the absolute highest point visible. Is the student correct?

Yes — the relative maximum must be the highest point on the entire graph.

Partially — $(6, 20)$ is a relative maximum, but $(2, 15)$ is also a relative maximum because it is higher than all nearby points, even though it is not the highest overall.

No — neither point is a relative maximum because the function has a global maximum somewhere else.

No — a graph can have at most one relative maximum.

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Exercises: Interpret Key Features of Graphs

Recall / Warm-Up

Fluency Practice