Fit Functions to Real-World Data - Statistics and Probability

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

24. Which model is suggested when first differences are constant?

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

25. What is the vertex form of a quadratic model?

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

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

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

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

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

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

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

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

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

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

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

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

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

32. Which model is suggested when second differences are constant?

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

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

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

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

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

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

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

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

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

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

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

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

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

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