Comparing Functions Represented in Different Ways - Algebra
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Which property is found by evaluating $f(0)$ for any function $f$?
Which property is found by evaluating $f(0)$ for any function $f$?
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The $y$-intercept. Substituting zero for x gives the y-coordinate where graph crosses y-axis.
The $y$-intercept. Substituting zero for x gives the y-coordinate where graph crosses y-axis.
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If $a>0$ in $f(x)=a(x-h)^2+k$, does the parabola have a max or min?
If $a>0$ in $f(x)=a(x-h)^2+k$, does the parabola have a max or min?
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Minimum. Positive coefficient means parabola opens upward with minimum.
Minimum. Positive coefficient means parabola opens upward with minimum.
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Which has smaller slope: table points $(1,6)$ and $(3,2)$ or $g(x)=-1x+0$?
Which has smaller slope: table points $(1,6)$ and $(3,2)$ or $g(x)=-1x+0$?
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The table function. Table slope is $(2-6)/(3-1)=-2$, which is less than $-1$.
The table function. Table slope is $(2-6)/(3-1)=-2$, which is less than $-1$.
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What is the range of $f(x)=a(x-h)^2+k$ when $a<0$?
What is the range of $f(x)=a(x-h)^2+k$ when $a<0$?
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$y\le k$. Downward parabola has maximum at vertex, range ends at k.
$y\le k$. Downward parabola has maximum at vertex, range ends at k.
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What is the range of $f(x)=a(x-h)^2+k$ when $a>0$?
What is the range of $f(x)=a(x-h)^2+k$ when $a>0$?
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$y\ge k$. Upward parabola has minimum at vertex, range starts at k.
$y\ge k$. Upward parabola has minimum at vertex, range starts at k.
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If $a<0$ in $f(x)=a(x-h)^2+k$, does the parabola have a max or min?
If $a<0$ in $f(x)=a(x-h)^2+k$, does the parabola have a max or min?
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Maximum. Negative coefficient means parabola opens downward with maximum.
Maximum. Negative coefficient means parabola opens downward with maximum.
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For $f(x)=a(x-h)^2+k$, what is the maximum or minimum value?
For $f(x)=a(x-h)^2+k$, what is the maximum or minimum value?
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$k$. The k-value represents the function's output at the vertex.
$k$. The k-value represents the function's output at the vertex.
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What is the vertex of $f(x)=a(x-h)^2+k$?
What is the vertex of $f(x)=a(x-h)^2+k$?
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$(h,k)$. Vertex form directly reveals the vertex coordinates.
$(h,k)$. Vertex form directly reveals the vertex coordinates.
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What is the axis of symmetry of $f(x)=a(x-h)^2+k$?
What is the axis of symmetry of $f(x)=a(x-h)^2+k$?
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$x=h$. Vertex form shows the axis of symmetry passes through the vertex.
$x=h$. Vertex form shows the axis of symmetry passes through the vertex.
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What is the slope of $f(x)=ax+b$ in terms of $a$?
What is the slope of $f(x)=ax+b$ in terms of $a$?
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$a$. In slope-intercept form, the coefficient of x is the slope.
$a$. In slope-intercept form, the coefficient of x is the slope.
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What is the y-intercept of $f(x)=ax+b$ in terms of $b$?
What is the y-intercept of $f(x)=ax+b$ in terms of $b$?
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$b$. In slope-intercept form, the constant term is the y-intercept.
$b$. In slope-intercept form, the constant term is the y-intercept.
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Identify which has smaller vertex $y$-value: $f(x)=- (x-2)^2+5$ or $g(x)=-(x-2)^2+1$.
Identify which has smaller vertex $y$-value: $f(x)=- (x-2)^2+5$ or $g(x)=-(x-2)^2+1$.
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$g(x)$. Compare vertex y-values: $1<5$.
$g(x)$. Compare vertex y-values: $1<5$.
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Identify which has greater vertex $y$-value: $f(x)=(x-1)^2+2$ or $g(x)=(x+3)^2-1$.
Identify which has greater vertex $y$-value: $f(x)=(x-1)^2+2$ or $g(x)=(x+3)^2-1$.
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$f(x)$. Compare vertex y-values: $2>-1$.
$f(x)$. Compare vertex y-values: $2>-1$.
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Which is increasing on all real numbers: $f(x)=5x-2$ or $g(x)=-x+7$?
Which is increasing on all real numbers: $f(x)=5x-2$ or $g(x)=-x+7$?
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$f(x)$. Positive slope means function increases everywhere.
$f(x)$. Positive slope means function increases everywhere.
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Which has the greater slope: $f(x)=-3x+2$ or $g(x)=\frac{1}{2}x-4$?
Which has the greater slope: $f(x)=-3x+2$ or $g(x)=\frac{1}{2}x-4$?
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$g(x)$. Compare slopes: $-3$ vs $\frac{1}{2}$, so $\frac{1}{2}>-3$.
$g(x)$. Compare slopes: $-3$ vs $\frac{1}{2}$, so $\frac{1}{2}>-3$.
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What key feature determines whether a quadratic has a larger maximum than another?
What key feature determines whether a quadratic has a larger maximum than another?
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Compare vertex $y$-values when $a<0$. Downward parabolas achieve maximum at their vertex point.
Compare vertex $y$-values when $a<0$. Downward parabolas achieve maximum at their vertex point.
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Identify the domain of a quadratic polynomial function in Algebra $1$.
Identify the domain of a quadratic polynomial function in Algebra $1$.
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All real numbers. Polynomial functions accept any real number as input.
All real numbers. Polynomial functions accept any real number as input.
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What is the slope between points $(x_1,y_1)$ and $(x_2,y_2)$?
What is the slope between points $(x_1,y_1)$ and $(x_2,y_2)$?
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$\frac{y_2-y_1}{x_2-x_1}$. Rise over run formula for finding rate of change.
$\frac{y_2-y_1}{x_2-x_1}$. Rise over run formula for finding rate of change.
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In a table, what pattern suggests a quadratic function instead of linear?
In a table, what pattern suggests a quadratic function instead of linear?
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Constant second differences. Quadratics have changing first differences but constant second differences.
Constant second differences. Quadratics have changing first differences but constant second differences.
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In a table, how do you identify a constant rate of change?
In a table, how do you identify a constant rate of change?
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Equal $\Delta y$ for equal $\Delta x$. Linear functions have constant rate of change between points.
Equal $\Delta y$ for equal $\Delta x$. Linear functions have constant rate of change between points.
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What does it mean if a function is decreasing on an interval?
What does it mean if a function is decreasing on an interval?
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As $x$ increases, $f(x)$ decreases. Function values fall as x-values move from left to right.
As $x$ increases, $f(x)$ decreases. Function values fall as x-values move from left to right.
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Which has larger value at $x=-2$: table gives $f(-2)=1$ or $g(x)=x^2$?
Which has larger value at $x=-2$: table gives $f(-2)=1$ or $g(x)=x^2$?
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$g(x)$. Compare values: $1<4$.
$g(x)$. Compare values: $1<4$.
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Which has greater slope: table points $(0,1)$ and $(2,9)$ or $g(x)=3x-4$?
Which has greater slope: table points $(0,1)$ and $(2,9)$ or $g(x)=3x-4$?
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The table function. Table slope is $(9-1)/(2-0)=4$, which is greater than $3$.
The table function. Table slope is $(9-1)/(2-0)=4$, which is greater than $3$.
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Which has smaller minimum: verbal “minimum value $-5$” or $g(x)=(x+2)^2-3$?
Which has smaller minimum: verbal “minimum value $-5$” or $g(x)=(x+2)^2-3$?
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The verbal function. Compare minimum values: $-5<-3$.
The verbal function. Compare minimum values: $-5<-3$.
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Which has larger maximum: verbal “maximum value $8$” or $g(x)=-(x-1)^2+6$?
Which has larger maximum: verbal “maximum value $8$” or $g(x)=-(x-1)^2+6$?
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The verbal function. Compare maximum values: $8>6$.
The verbal function. Compare maximum values: $8>6$.
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What does it mean if a function is increasing on an interval?
What does it mean if a function is increasing on an interval?
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As $x$ increases, $f(x)$ increases. Function values rise as x-values move from left to right.
As $x$ increases, $f(x)$ increases. Function values rise as x-values move from left to right.
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Which has larger initial value: table shows $f(0)=-2$ or algebraic $g(x)=-5x+1$?
Which has larger initial value: table shows $f(0)=-2$ or algebraic $g(x)=-5x+1$?
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$g(x)$. Compare initial values: $-2<1$, so $g(x)$ is larger.
$g(x)$. Compare initial values: $-2<1$, so $g(x)$ is larger.
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Which has larger initial value: verbal $f(0)=7$ or algebraic $g(x)=2x+1$?
Which has larger initial value: verbal $f(0)=7$ or algebraic $g(x)=2x+1$?
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$f$. Compare initial values: $7>1$.
$f$. Compare initial values: $7>1$.
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A function is described as “starts at $-3$ when $x=0$ and increases $\frac{1}{2}$ per $1$ unit.” What is $f(x)$?
A function is described as “starts at $-3$ when $x=0$ and increases $\frac{1}{2}$ per $1$ unit.” What is $f(x)$?
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$f(x)=\frac{1}{2}x-3$. Starts at $-3$ with slope $\frac{1}{2}$ gives this form.
$f(x)=\frac{1}{2}x-3$. Starts at $-3$ with slope $\frac{1}{2}$ gives this form.
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A function is described as “starts at $4$ when $x=0$ and decreases $2$ per $1$ unit.” What is the slope?
A function is described as “starts at $4$ when $x=0$ and decreases $2$ per $1$ unit.” What is the slope?
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$-2$. Decreases 2 per unit means slope is $-2$.
$-2$. Decreases 2 per unit means slope is $-2$.
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