Fit Functions to Real-World Data - Statistics
Card 1 of 30
What condition on $b$ in $y=a\cdot b^x$ indicates exponential decay?
What condition on $b$ in $y=a\cdot b^x$ indicates exponential decay?
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$0<b<1$. Values between 0 and 1 represent decay (decreasing function).
$0<b<1$. Values between 0 and 1 represent decay (decreasing function).
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What is the standard form of an exponential model (with initial value $a$)?
What is the standard form of an exponential model (with initial value $a$)?
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$y=a\cdot b^x$. $a$ is initial value when $x=0$, $b$ is growth/decay factor.
$y=a\cdot b^x$. $a$ is initial value when $x=0$, $b$ is growth/decay factor.
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What is the vertex form of a quadratic model?
What is the vertex form of a quadratic model?
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$y=a(x-h)^2+k$. Parabola with vertex at $(h,k)$ and vertical stretch factor $a$.
$y=a(x-h)^2+k$. Parabola with vertex at $(h,k)$ and vertical stretch factor $a$.
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What is the slope-intercept form of a linear model?
What is the slope-intercept form of a linear model?
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$y=mx+b$. $m$ is slope, $b$ is $y$-intercept where line crosses $y$-axis.
$y=mx+b$. $m$ is slope, $b$ is $y$-intercept where line crosses $y$-axis.
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What is the slope formula between two points $(x_1,y_1)$ and $(x_2,y_2)$?
What is the slope formula between two points $(x_1,y_1)$ and $(x_2,y_2)$?
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$m=\frac{y_2-y_1}{x_2-x_1}$. Rise over run: change in $y$ divided by change in $x$.
$m=\frac{y_2-y_1}{x_2-x_1}$. Rise over run: change in $y$ divided by change in $x$.
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Which model type is suggested by a constant percent change (constant ratio) per equal $x$ step?
Which model type is suggested by a constant percent change (constant ratio) per equal $x$ step?
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Exponential model. Constant ratio between consecutive $y$ values indicates exponential growth/decay.
Exponential model. Constant ratio between consecutive $y$ values indicates exponential growth/decay.
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Which model type is suggested when second differences are constant across equal $x$ steps?
Which model type is suggested when second differences are constant across equal $x$ steps?
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Quadratic model. Second differences are differences of first differences; constant indicates parabolic shape.
Quadratic model. Second differences are differences of first differences; constant indicates parabolic shape.
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Which model type is suggested when first differences are constant across equal $x$ steps?
Which model type is suggested when first differences are constant across equal $x$ steps?
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Linear model. First differences are $y_{i+1}-y_i$; constant means linear relationship.
Linear model. First differences are $y_{i+1}-y_i$; constant means linear relationship.
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Evaluate the exponential model $y=50(0.8)^x$ at $x=2$.
Evaluate the exponential model $y=50(0.8)^x$ at $x=2$.
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$y=32$. $y=50(0.8)^2=50(0.64)=32$.
$y=32$. $y=50(0.8)^2=50(0.64)=32$.
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Use $y=2x+1$ to predict $y$ when $x=7$.
Use $y=2x+1$ to predict $y$ when $x=7$.
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$y=15$. Substitute: $y=2(7)+1=14+1=15$.
$y=15$. Substitute: $y=2(7)+1=14+1=15$.
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In $y=3(x-2)^2-5$, what is the vertex $(h,k)$?
In $y=3(x-2)^2-5$, what is the vertex $(h,k)$?
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$(2,-5)$. Vertex form shows vertex directly: $h=2$, $k=-5$.
$(2,-5)$. Vertex form shows vertex directly: $h=2$, $k=-5$.
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Which model type is suggested by a constant rate of change in a scatter plot?
Which model type is suggested by a constant rate of change in a scatter plot?
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Linear model. Constant rate of change means equal $y$ changes for equal $x$ changes.
Linear model. Constant rate of change means equal $y$ changes for equal $x$ changes.
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Find the linear equation through $(0,3)$ with slope $-4$.
Find the linear equation through $(0,3)$ with slope $-4$.
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$y=-4x+3$. Point $(0,3)$ gives $y$-intercept $b=3$; slope $m=-4$.
$y=-4x+3$. Point $(0,3)$ gives $y$-intercept $b=3$; slope $m=-4$.
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Find the slope of the line through $(2,5)$ and $(6,13)$.
Find the slope of the line through $(2,5)$ and $(6,13)$.
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$m=2$. $m=\frac{13-5}{6-2}=\frac{8}{4}=2$.
$m=2$. $m=\frac{13-5}{6-2}=\frac{8}{4}=2$.
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Identify the model: $y=4.5x-11$ (choose linear, quadratic, or exponential).
Identify the model: $y=4.5x-11$ (choose linear, quadratic, or exponential).
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Linear. Form $y=mx+b$ with constant slope.
Linear. Form $y=mx+b$ with constant slope.
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Identify the model: $y=-2(x-3)^2+7$ (choose linear, quadratic, or exponential).
Identify the model: $y=-2(x-3)^2+7$ (choose linear, quadratic, or exponential).
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Quadratic. Vertex form $y=a(x-h)^2+k$ indicates parabola.
Quadratic. Vertex form $y=a(x-h)^2+k$ indicates parabola.
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Use $y=100(1.2)^t$ to predict $y$ when $t=3$.
Use $y=100(1.2)^t$ to predict $y$ when $t=3$.
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$y=172.8$. $y=100(1.2)^3=100(1.728)=172.8$.
$y=172.8$. $y=100(1.2)^3=100(1.728)=172.8$.
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Find the percent increase per $1$ unit of $x$ for $y=80(1.06)^x$.
Find the percent increase per $1$ unit of $x$ for $y=80(1.06)^x$.
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$6%$ increase per unit $x$. $b=1.06$ means multiply by $1.06$ (6% increase) per unit.
$6%$ increase per unit $x$. $b=1.06$ means multiply by $1.06$ (6% increase) per unit.
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Identify the model: $y=120(1.05)^t$ (choose linear, quadratic, or exponential).
Identify the model: $y=120(1.05)^t$ (choose linear, quadratic, or exponential).
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Exponential. Form $y=a\cdot b^x$ with $a=120$, $b=1.05$.
Exponential. Form $y=a\cdot b^x$ with $a=120$, $b=1.05$.
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What does the parameter $b$ represent in $y=a\cdot b^x$?
What does the parameter $b$ represent in $y=a\cdot b^x$?
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Growth/decay factor per $1$ unit increase in $x$. Each unit increase in $x$ multiplies $y$ by factor $b$.
Growth/decay factor per $1$ unit increase in $x$. Each unit increase in $x$ multiplies $y$ by factor $b$.
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Find the vertex $x$-coordinate of $y = 2x^2 - 8x + 1$.
Find the vertex $x$-coordinate of $y = 2x^2 - 8x + 1$.
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$x = 2$. Use vertex formula: $x = -\frac{-8}{2(2)} = \frac{8}{4} = 2$.
$x = 2$. Use vertex formula: $x = -\frac{-8}{2(2)} = \frac{8}{4} = 2$.
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Given $(0,5)$ and $(2,20)$ fit $y = a\cdot b^x$. What is $b$?
Given $(0,5)$ and $(2,20)$ fit $y = a\cdot b^x$. What is $b$?
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$b = 2$. $\frac{20}{5} = 4 = 2^2$, so $b = 2$.
$b = 2$. $\frac{20}{5} = 4 = 2^2$, so $b = 2$.
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What does the sign of $a$ indicate in $y = ax^2 + bx + c$?
What does the sign of $a$ indicate in $y = ax^2 + bx + c$?
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$a>0$ opens up; $a<0$ opens down. The coefficient of $x^2$ determines the parabola's direction.
$a>0$ opens up; $a<0$ opens down. The coefficient of $x^2$ determines the parabola's direction.
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Identify the model type: $y = -0.5x^2 + 3x + 2$.
Identify the model type: $y = -0.5x^2 + 3x + 2$.
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Quadratic. The $x^2$ term indicates a quadratic function.
Quadratic. The $x^2$ term indicates a quadratic function.
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What is the vertex $x$-coordinate of $y = ax^2 + bx + c$?
What is the vertex $x$-coordinate of $y = ax^2 + bx + c$?
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$x = -\frac{b}{2a}$. The vertex formula locates the parabola's turning point.
$x = -\frac{b}{2a}$. The vertex formula locates the parabola's turning point.
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What does the parameter $b$ represent in $y = mx + b$ in context?
What does the parameter $b$ represent in $y = mx + b$ in context?
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Predicted value of $y$ when $x = 0$. The $y$-intercept is the starting value in many contexts.
Predicted value of $y$ when $x = 0$. The $y$-intercept is the starting value in many contexts.
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What does the slope $m$ represent in a linear model $y = mx + b$ in context?
What does the slope $m$ represent in a linear model $y = mx + b$ in context?
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Change in $y$ per $1$ unit increase in $x$. Slope measures the rate of change in the dependent variable.
Change in $y$ per $1$ unit increase in $x$. Slope measures the rate of change in the dependent variable.
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What is the $y$-intercept of $y = -3x + 7$?
What is the $y$-intercept of $y = -3x + 7$?
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$7$. The constant term is the $y$-intercept when $x = 0$.
$7$. The constant term is the $y$-intercept when $x = 0$.
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Find the linear equation through $(0,3)$ and $(4,11)$.
Find the linear equation through $(0,3)$ and $(4,11)$.
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$y = 2x + 3$. Slope is $\frac{11-3}{4-0} = 2$, and $y$-intercept is $3$.
$y = 2x + 3$. Slope is $\frac{11-3}{4-0} = 2$, and $y$-intercept is $3$.
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What is the slope of the line through $(2,5)$ and $(6,13)$?
What is the slope of the line through $(2,5)$ and $(6,13)$?
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$m = 2$. Apply slope formula: $m = \frac{13-5}{6-2} = \frac{8}{4} = 2$.
$m = 2$. Apply slope formula: $m = \frac{13-5}{6-2} = \frac{8}{4} = 2$.
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