Interpretting Functions: Interpreting and Sketching Key Features of Functions (CCSS.F-IF.4)
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Common Core High School Functions › Interpretting Functions: Interpreting and Sketching Key Features of Functions (CCSS.F-IF.4)
A function $f$ is sampled at these inputs:
x: -3, -2, -1, 0, 1, 2, 3 f(x): 5, 3, 1, 0, 2, 4, 6
Which statement about the function is true?
The function is positive for all shown $x$-values.
The function has a relative minimum at $x=0$.
The function is increasing on $[-3,-1]$.
The function has a zero at $x=1$.
Explanation
From $x=-3$ to $x=0$, $f$ decreases (5, 3, 1, 0), and from $x=0$ to $x=3$, $f$ increases (0, 2, 4, 6). Thus $x=0$ is a turning point with lower value than its neighbors, so there is a relative minimum at $x=0$. A is false because $f(0)=0$ (not positive). C is false because the values decrease on $[-3,-1]$. D is false since $f(1)=2$.
A function $g$ is sampled at these inputs:
x: 0, 2, 4, 6, 8 g(x): 7, 5, 5, 4, 3
Which statement about the function is true?
The function is increasing on $[2,6]$.
The function has a relative maximum at $x=6$.
$g(4) < g(6)$.
The function is non-increasing on $[0,8]$.
Explanation
From $x=0$ to $2$, $g$ decreases (7 to 5); from $2$ to $4$, it is constant (5 to 5); from $4$ to $8$, it decreases (5 to 3). That is non-increasing over $[0,8]$. A is false (not increasing on $[2,6]$ because it is constant then decreases). B is false since $g(6)=4$ is not larger than neighbors. C is false because $g(4)=5$ is not less than $g(6)=4$.
A function $h$ is sampled at these inputs:
x: -4, -2, 0, 2, 4 h(x): -3, -1, 2, 1, -2
Which statement about the function is true?
The function changes sign between $x=-2$ and $x=0$.
The function is positive for all $x>0$ shown.
The function is negative for all shown $x$-values.
The function has a zero at $x=2$.
Explanation
The values go from $h(-2)=-1$ (negative) to $h(0)=2$ (positive), so there is a sign change between $-2$ and $0$. B is false because $h(4)=-2$. C is false because $h(0)=2$ and $h(2)=1$ are positive. D is false since $h(2)=1$, not zero.
A function $p$ is sampled at these inputs:
x: -3, -2, -1, 0, 1, 2, 3 p(x): 10, 5, 2, 1, 2, 5, 10
Which statement about the function is true?
The function appears to be odd: $p(-x)=-p(x)$ for the shown values.
The function is decreasing for all shown $x$-values.
The function appears to be even: $p(-x)=p(x)$ for the shown values.
The function has a relative minimum at $x=2$.
Explanation
For each pair of opposites, $p(-3)=p(3)=10$, $p(-2)=p(2)=5$, $p(-1)=p(1)=2$, and $p(0)=1$, indicating even symmetry on the sampled points. A is false (values are not negatives). B is false as values increase from $x=0$ to $x=3$. D is false because $p(2)=5$ is not less than both neighbors (it is greater than $p(1)=2$ and less than $p(3)=10$).
A function $q$ is sampled at these inputs:
x: -1, 0, 1, 2, 3, 4 q(x): 4, 3, 1, 1, 2, 5
Which statement about the function is true?
The function is decreasing on $[-1,1]$.
$q(2) < q(1)$.
The function is constant on $[2,4]$.
The function is negative for $x\ge 3$.
Explanation
From $x=-1$ to $x=1$, the values go 4, 3, 1, which is strictly decreasing. B is false because $q(2)=1$ equals $q(1)=1$. C is false since values on $[2,4]$ are 1, 2, 5 (not constant). D is false because $q(3)=2$ and $q(4)=5$ are positive.
Values of a function are shown: $x$: -2, -1, 0, 1, 2 $f(x)$: -3, -1, 1, 3, 5
Which statement about the function is true?
The function has a relative maximum at $x=-1$.
The function is increasing on the entire interval shown.
The function is negative for all $x$ shown.
$f(0) < f(-1)$.
Explanation
Each time $x$ increases by 1, $f(x)$ increases: $-3,-1,1,3,5$, so the function is increasing on the interval shown. There is no relative maximum at $x=-1$ (the values continue to rise), the function is not negative for all $x$ (e.g., $f(0)=1$), and $f(0)=1$ is not less than $f(-1)=-1$.
Values of a function are shown: $x$: -3, -2, -1, 0, 1, 2 $f(x)$: 5, 2, 0, 1, 4, 10
Which statement about the function is true?
$f$ is increasing on $[-3,-1]$.
$f$ is negative for all $x$ shown.
$f$ has a relative maximum at $x=-1$.
$f$ has a relative minimum at $x=-1$.
Explanation
From $x=-3$ to $x=-1$, $f(x)$ decreases (5 → 2 → 0), then from $x=-1$ onward it increases (0 → 1 → 4 → 10). Thus $x=-1$ is a relative minimum. The other statements either reverse the direction, claim all values are negative (they are not), or call $x=-1$ a maximum (it is the smallest shown).
Values of a function are shown: $x$: 0, 1, 2, 3, 4 $f(x)$: 8, 5, 2, -1, -4
Which statement about the function is true?
$f$ is decreasing on the interval shown.
$f(4) > f(0)$.
$f$ is positive for all $x$ shown.
$f$ has a relative minimum at $x=2$.
Explanation
As $x$ increases, $f(x)$ moves 8 → 5 → 2 → −1 → −4, so it decreases throughout. $f(4)$ is not greater than $f(0)$ (−4 < 8), the function is not always positive (negative at $x=3,4$), and there is no relative minimum at $x=2$ because the values continue to decrease after $x=2$.
Values of a function are shown: $x$: -3, -2, -1, 0, 1, 2, 3 $f(x)$: 10, 5, 2, 1, 2, 5, 10
Which statement about the function is true?
$f$ is odd: $f(-x)=-f(x)$.
$f$ is decreasing for all $x$ shown.
$f$ is symmetric about the $y$!-axis.
$f$ has a maximum at $x=0$.
Explanation
The data satisfy $f(-x)=f(x)$ (e.g., $f(-2)=5=f(2)$), indicating symmetry about the $y$-axis (an even function). It is not always decreasing (it decreases to $x=0$ then increases), it is not odd, and $x=0$ is where the minimum occurs, not a maximum.
A function is given by the points: $(-2,3)$, $(-1,1)$, $(0,2)$, $(1,4)$, $(2,1)$.
Which statement about the function is true?
The maximum value shown is 4 at $x=1$.
The function is increasing on the entire interval from $x=-2$ to $x=2$.
$f(2) > f(1)$.
The function is negative at $x=-2$.
Explanation
From the listed values, the largest $f(x)$ is 4 at $x=1$. The function is not increasing over the entire interval (it drops from 4 at $x=1$ to 1 at $x=2$), $f(2)=1$ is not greater than $f(1)=4$, and $f(-2)=3$ is not negative.