Interpreting/Sketching Key Features of Functions - Algebra 2
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What is the slope interpretation of average rate of change on $[a,b]$?
What is the slope interpretation of average rate of change on $[a,b]$?
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It is the slope of the secant line: $\frac{f(b)-f(a)}{b-a}$. Geometric interpretation of average rate of change.
It is the slope of the secant line: $\frac{f(b)-f(a)}{b-a}$. Geometric interpretation of average rate of change.
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What is the definition of the $x$-intercept of a function’s graph?
What is the definition of the $x$-intercept of a function’s graph?
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A point where $y=0$ on the graph, so $f(x)=0$. Where the graph crosses the $x$-axis.
A point where $y=0$ on the graph, so $f(x)=0$. Where the graph crosses the $x$-axis.
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Which key feature is identified by evaluating $f(0)$?
Which key feature is identified by evaluating $f(0)$?
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The $y$-intercept $(0,f(0))$. Found by evaluating the function at zero.
The $y$-intercept $(0,f(0))$. Found by evaluating the function at zero.
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Which key feature is identified by solving $f(x)=0$?
Which key feature is identified by solving $f(x)=0$?
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The $x$-intercepts (zeros) of the function. Found by setting the function equal to zero.
The $x$-intercepts (zeros) of the function. Found by setting the function equal to zero.
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What does the statement “$f$ is increasing on $(a,b)$” mean using inequalities?
What does the statement “$f$ is increasing on $(a,b)$” mean using inequalities?
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If $x_1<x_2$ in $(a,b)$, then $f(x_1)<f(x_2)$. Definition using ordered pairs in the interval.
If $x_1<x_2$ in $(a,b)$, then $f(x_1)<f(x_2)$. Definition using ordered pairs in the interval.
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What does the statement “$f$ is positive on $(a,b)$” mean?
What does the statement “$f$ is positive on $(a,b)$” mean?
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For all $x$ in $(a,b)$, $f(x)>0$. All function values are above the $x$-axis.
For all $x$ in $(a,b)$, $f(x)>0$. All function values are above the $x$-axis.
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Identify the symmetry of $f(x)=\frac{1}{x}$.
Identify the symmetry of $f(x)=\frac{1}{x}$.
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Odd; it has origin symmetry. Standard reciprocal function has origin symmetry.
Odd; it has origin symmetry. Standard reciprocal function has origin symmetry.
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What is the end behavior of $f(x)=\frac{1}{x-3}$ as $x\to\pm\infty$?
What is the end behavior of $f(x)=\frac{1}{x-3}$ as $x\to\pm\infty$?
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As $x\to\pm\infty$, $f(x)\to 0$. Rational function approaches horizontal asymptote.
As $x\to\pm\infty$, $f(x)\to 0$. Rational function approaches horizontal asymptote.
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Identify the horizontal asymptote of $f(x)=\frac{1}{x-3}$.
Identify the horizontal asymptote of $f(x)=\frac{1}{x-3}$.
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Horizontal asymptote: $y=0$. Rational function approaches zero as $x$ grows.
Horizontal asymptote: $y=0$. Rational function approaches zero as $x$ grows.
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Identify the vertical asymptote of $f(x)=\frac{1}{x-3}$.
Identify the vertical asymptote of $f(x)=\frac{1}{x-3}$.
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Vertical asymptote: $x=3$. Denominator cannot equal zero.
Vertical asymptote: $x=3$. Denominator cannot equal zero.
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Identify the domain restriction implied by the model $f(x)=\sqrt{x-4}$.
Identify the domain restriction implied by the model $f(x)=\sqrt{x-4}$.
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Domain: $x\ge 4$. Expression under square root must be non-negative.
Domain: $x\ge 4$. Expression under square root must be non-negative.
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Which value is the $y$-intercept of $f(x)=\sqrt{x+1}$?
Which value is the $y$-intercept of $f(x)=\sqrt{x+1}$?
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The $y$-intercept is $(0,1)$. Evaluate $f(0)=\sqrt{0+1}=1$.
The $y$-intercept is $(0,1)$. Evaluate $f(0)=\sqrt{0+1}=1$.
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Identify the open intervals where $f(x)=(x-2)(x-5)$ is positive.
Identify the open intervals where $f(x)=(x-2)(x-5)$ is positive.
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$(-\infty,2)\cup(5,\infty)$. Parabola is positive outside its roots.
$(-\infty,2)\cup(5,\infty)$. Parabola is positive outside its roots.
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Identify the open interval where $f(x)=(x-2)(x-5)$ is negative.
Identify the open interval where $f(x)=(x-2)(x-5)$ is negative.
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$ (2,5)$. Parabola is negative between its roots.
$ (2,5)$. Parabola is negative between its roots.
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Identify the average rate of change from $x=2$ to $x=5$ given $f(2)=1$ and $f(5)=7$.
Identify the average rate of change from $x=2$ to $x=5$ given $f(2)=1$ and $f(5)=7$.
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Average rate of change is $\frac{7-1}{5-2}=2$. Formula: $\frac{\text{change in output}}{\text{change in input}}$.
Average rate of change is $\frac{7-1}{5-2}=2$. Formula: $\frac{\text{change in output}}{\text{change in input}}$.
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What does the point $(a,b)$ in a table for $f$ represent in context?
What does the point $(a,b)$ in a table for $f$ represent in context?
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When input is $a$, the output is $f(a)=b$. Input-output relationship in tabular form.
When input is $a$, the output is $f(a)=b$. Input-output relationship in tabular form.
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Identify the period of $y=\cos(\frac{x}{3})$.
Identify the period of $y=\cos(\frac{x}{3})$.
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The period is $6\pi$. Period formula: $\frac{2\pi}{1/3}=6\pi$.
The period is $6\pi$. Period formula: $\frac{2\pi}{1/3}=6\pi$.
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Identify the period of $y=\sin(4x)$.
Identify the period of $y=\sin(4x)$.
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The period is $\frac{\pi}{2}$. Period formula: $\frac{2\pi}{4}=\frac{\pi}{2}$.
The period is $\frac{\pi}{2}$. Period formula: $\frac{2\pi}{4}=\frac{\pi}{2}$.
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What is the midline of $y=3\sin(x)+2$?
What is the midline of $y=3\sin(x)+2$?
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The midline is $y=2$. Vertical shift of the sine function.
The midline is $y=2$. Vertical shift of the sine function.
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What is the amplitude of $y=3\sin(x)$?
What is the amplitude of $y=3\sin(x)$?
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The amplitude is $3$. Coefficient of sine function.
The amplitude is $3$. Coefficient of sine function.
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Identify whether $f(x)=x^2+x$ is even, odd, or neither.
Identify whether $f(x)=x^2+x$ is even, odd, or neither.
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Neither. Neither even nor odd function.
Neither. Neither even nor odd function.
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Which statement gives the end behavior of $f(x)=-3x^3+2$?
Which statement gives the end behavior of $f(x)=-3x^3+2$?
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As $x\to\infty$, $f(x)\to-\infty$; as $x\to-\infty$, $f(x)\to\infty$. Odd degree with negative leading coefficient.
As $x\to\infty$, $f(x)\to-\infty$; as $x\to-\infty$, $f(x)\to\infty$. Odd degree with negative leading coefficient.
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Which statement gives the end behavior of $f(x)=x^4-2x^2$?
Which statement gives the end behavior of $f(x)=x^4-2x^2$?
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As $x\to\pm\infty$, $f(x)\to\infty$. Even degree with positive leading coefficient.
As $x\to\pm\infty$, $f(x)\to\infty$. Even degree with positive leading coefficient.
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Identify the vertex of $f(x)=x^2-6x+10$.
Identify the vertex of $f(x)=x^2-6x+10$.
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The vertex is $(3,1)$. From vertex form $(x-3)^2+1$.
The vertex is $(3,1)$. From vertex form $(x-3)^2+1$.
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Identify the axis of symmetry of $f(x)=x^2-6x+10$.
Identify the axis of symmetry of $f(x)=x^2-6x+10$.
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The axis of symmetry is $x=3$. Complete the square: $(x-3)^2+1$.
The axis of symmetry is $x=3$. Complete the square: $(x-3)^2+1$.
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On which interval is $f(x)=-2(x+1)^2+7$ decreasing?
On which interval is $f(x)=-2(x+1)^2+7$ decreasing?
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Decreasing on $(-1,\infty)$. Right side of downward-opening parabola.
Decreasing on $(-1,\infty)$. Right side of downward-opening parabola.
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On which interval is $f(x)=-2(x+1)^2+7$ increasing?
On which interval is $f(x)=-2(x+1)^2+7$ increasing?
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Increasing on $(-\infty,-1)$. Left side of downward-opening parabola.
Increasing on $(-\infty,-1)$. Left side of downward-opening parabola.
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On which interval is $f(x)=(x-3)^2+5$ increasing?
On which interval is $f(x)=(x-3)^2+5$ increasing?
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Increasing on $(3,\infty)$. Right side of upward-opening parabola.
Increasing on $(3,\infty)$. Right side of upward-opening parabola.
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On which interval is $f(x)=(x-3)^2+5$ decreasing?
On which interval is $f(x)=(x-3)^2+5$ decreasing?
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Decreasing on $(-\infty,3)$. Left side of upward-opening parabola.
Decreasing on $(-\infty,3)$. Left side of upward-opening parabola.
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Identify the relative maximum point of $f(x)=-2(x+1)^2+7$.
Identify the relative maximum point of $f(x)=-2(x+1)^2+7$.
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The relative maximum is at $(-1,7)$. Parabola opens downward with vertex at $(-1,7)$.
The relative maximum is at $(-1,7)$. Parabola opens downward with vertex at $(-1,7)$.
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