Statistics Flashcards: Fit Functions To Real World Data

What this deck covers

This deck focuses on Fit Functions To Real World Data, giving you a quick way to review the definitions, rules, and examples that matter most for Statistics.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: Identify the model: $y=-2(x-3)^2+7$ (choose linear, quadratic, or exponential).

Answer: Quadratic. Vertex form $y=a(x-h)^2+k$ indicates parabola.

Flashcard 2: Identify the model: $y=120(1.05)^t$ (choose linear, quadratic, or exponential).

Answer: Exponential. Form $y=a\cdot b^x$ with $a=120$, $b=1.05$.

Flashcard 3: Find the vertex $x$-coordinate of $y = 2x^2 - 8x + 1$.

Answer: $x = 2$. Use vertex formula: $x = -\frac{-8}{2(2)} = \frac{8}{4} = 2$.

Flashcard 4: Which model type is suggested by a constant rate of change in a scatter plot?

Answer: Linear model. Constant rate of change means equal $y$ changes for equal $x$ changes.

Flashcard 5: In $y=3(x-2)^2-5$, what is the vertex $(h,k)$?

Answer: $(2,-5)$. Vertex form shows vertex directly: $h=2$, $k=-5$.

Flashcard 6: Find the linear equation through $(0,3)$ and $(4,11)$.

Answer: $y = 2x + 3$. Slope is $\frac{11-3}{4-0} = 2$, and $y$-intercept is $3$.

Flashcard 7: Identify the model type: $y = -0.5x^2 + 3x + 2$.

Answer: Quadratic. The $x^2$ term indicates a quadratic function.

Flashcard 8: Evaluate the exponential model $y=50(0.8)^x$ at $x=2$.

Answer: $y=32$. $y=50(0.8)^2=50(0.64)=32$.

Flashcard 9: Given $(0,5)$ and $(2,20)$ fit $y = a\cdot b^x$. What is $a$?

Answer: $a = 5$. When $x = 0$, $y = a \cdot b^0 = a \cdot 1 = a$.

Flashcard 10: Which model type is suggested when second differences are constant across equal $x$ steps?

Answer: Quadratic model. Second differences are differences of first differences; constant indicates parabolic shape.

Flashcard 11: What condition on $b$ in $y=a\cdot b^x$ indicates exponential decay?

Answer: $0<b<1$. Values between 0 and 1 represent decay (decreasing function).

Flashcard 12: What is the slope of the line through $(2,5)$ and $(6,13)$?

Answer: $m = 2$. Apply slope formula: $m = \frac{13-5}{6-2} = \frac{8}{4} = 2$.

Flashcard 13: Identify the model: $y=4.5x-11$ (choose linear, quadratic, or exponential).

Answer: Linear. Form $y=mx+b$ with constant slope.

Flashcard 14: Use $y = 3\cdot 2^x$ to predict $y$ when $x = 4$.

Answer: $y = 48$. Substitute: $y = 3 \cdot 2^4 = 3 \cdot 16 = 48$.

Flashcard 15: Which model is suggested by a constant rate of change in the data?

Answer: Linear model: $y = mx + b$. Constant rate of change means equal changes in $y$ for equal changes in $x$.

Flashcard 16: What does the slope $m$ represent in a linear model $y = mx + b$ in context?

Answer: Change in $y$ per $1$ unit increase in $x$. Slope measures the rate of change in the dependent variable.

Flashcard 17: Use $y=2x+1$ to predict $y$ when $x=7$.

Answer: $y=15$. Substitute: $y=2(7)+1=14+1=15$.

Flashcard 18: What does the sign of $a$ indicate in $y = ax^2 + bx + c$?

Answer: $a>0$ opens up; $a<0$ opens down. The coefficient of $x^2$ determines the parabola's direction.

Flashcard 19: What is the $y$-intercept of $y = -3x + 7$?

Answer: $7$. The constant term is the $y$-intercept when $x = 0$.

Flashcard 20: What is the standard form of an exponential model (with initial value $a$)?

Answer: $y=a\cdot b^x$. $a$ is initial value when $x=0$, $b$ is growth/decay factor.

Flashcard 21: Find the slope of the line through $(2,5)$ and $(6,13)$.

Answer: $m=2$. $m=\frac{13-5}{6-2}=\frac{8}{4}=2$.

Flashcard 22: What is the slope-intercept form of a linear model?

Answer: $y=mx+b$. $m$ is slope, $b$ is $y$-intercept where line crosses $y$-axis.

Flashcard 23: Find the percent increase per $1$ unit of $x$ for $y=80(1.06)^x$.

Answer: $6\%$ increase per unit $x$. $b=1.06$ means multiply by $1.06$ (6% increase) per unit.

Flashcard 24: Which model is suggested when first differences are constant?

Answer: Linear model. First differences are the changes between consecutive $y$-values.

Flashcard 25: What is the vertex form of a quadratic model?

Answer: $y=a(x-h)^2+k$. Parabola with vertex at $(h,k)$ and vertical stretch factor $a$.

Flashcard 26: Use $y=100(1.2)^t$ to predict $y$ when $t=3$.

Answer: $y=172.8$. $y=100(1.2)^3=100(1.728)=172.8$.

Flashcard 27: Which model is suggested when the data changes by a constant percent each step?

Answer: Exponential model: $y = a\cdot b^x$. Constant percent change means multiplying by the same factor each step.

Flashcard 28: Which model type is suggested when first differences are constant across equal $x$ steps?

Answer: Linear model. First differences are $y_{i+1}-y_i$; constant means linear relationship.

Flashcard 29: What does the parameter $b$ represent in $y=a\cdot b^x$?

Answer: Growth/decay factor per $1$ unit increase in $x$. Each unit increase in $x$ multiplies $y$ by factor $b$.

Flashcard 30: Find the linear equation through $(0,3)$ with slope $-4$.

Answer: $y=-4x+3$. Point $(0,3)$ gives $y$-intercept $b=3$; slope $m=-4$.

Flashcard 31: What is the residual for observed $y=18$ when the model predicts $\hat{y}=20$?

Answer: $-2$. Residual = observed - predicted = $18 - 20 = -2$.

Flashcard 32: Which model is suggested when second differences are constant?

Answer: Quadratic model: $y = ax^2 + bx + c$. Second differences are differences of first differences.

Flashcard 33: Given $(0,5)$ and $(2,20)$ fit $y = a\cdot b^x$. What is $b$?

Answer: $b = 2$. $\frac{20}{5} = 4 = 2^2$, so $b = 2$.

Flashcard 34: In $y = a\cdot b^x$, what condition indicates exponential growth versus decay?

Answer: Growth: $b>1$; decay: $0<b<1$. Base greater than 1 multiplies to increase; less than 1 to decrease.

Flashcard 35: What does the parameter $b$ represent in $y = mx + b$ in context?

Answer: Predicted value of $y$ when $x = 0$. The $y$-intercept is the starting value in many contexts.

Flashcard 36: What is the slope formula between two points $(x_1,y_1)$ and $(x_2,y_2)$?

Answer: $m=\frac{y_2-y_1}{x_2-x_1}$. Rise over run: change in $y$ divided by change in $x$.

Flashcard 37: What is the vertex $x$-coordinate of $y = ax^2 + bx + c$?

Answer: $x = -\frac{b}{2a}$. The vertex formula locates the parabola's turning point.

Flashcard 38: Which model type is suggested by a constant percent change (constant ratio) per equal $x$ step?

Answer: Exponential model. Constant ratio between consecutive $y$ values indicates exponential growth/decay.