Question

modeling dat
objectives:

1. graph exponential
2. use exponential
3. use logarithmic
4. graph logarithmic
5. use quadratic
6. graph quadratic
7. determine an

scatter plots and re

• a line that b
• for example
• represents a

modeling with ex
the expor
other tha
smple 1: gra

1. connect these points with a smooth curve.

example 6: graph the quadratic function: (y = x^{2}-2x - 3)
determine an appropriate function for modeling data:
description of data points in a scatter plot\tmodel
lie on or near a line\tlinear function (y=mx + b) or (f(x)=mx + b)
increasing more and more rapidly\texponential function (y = b^{x}), or (f(x)=b^{x},b > 1)
increasing, although rate of increase is slowing down\tlogarithmic function, (y=log_{b}x,b > 1)
(y = log_{b}x) means (b^{y}=x).
decreasing and then increasing\tquadratic function (y = ax^{2}+bx + c) or (f(x)=ax^{2}+bx + c,a>0). the vertex is a minimum.
increasing and then decreasing\tquadratic function (y = ax^{2}+bx + c) or (f(x)=ax^{2}+bx + c,a < 0). the vertex is a maximum.

Explanation:

Step1: Analyze the description of data points

The key is to match the data - point behavior with the function type.

Step2: For data points lying on or near a line

The appropriate function is a linear function $y = mx + b$ or $f(x)=mx + b$.

Step3: For data points increasing more and more rapidly

The appropriate function is an exponential function $y = b^{x}$ or $f(x)=b^{x},b > 1$.

Step4: For data points increasing with rate of increase slowing down

The appropriate function is a logarithmic function $y=\log_{b}x,b > 1$ where $y = \log_{b}x$ means $b^{y}=x$.

Step5: For data points decreasing and then increasing

The appropriate function is a quadratic function $y = ax^{2}+bx + c$ or $f(x)=ax^{2}+bx + c,a>0$ (vertex is a minimum).

Step6: For data points increasing and then decreasing

The appropriate function is a quadratic function $y = ax^{2}+bx + c$ or $f(x)=ax^{2}+bx + c,a < 0$ (vertex is a maximum).

Answer:

Based on the description of data - points in a scatter - plot, the appropriate functions are as follows:

• If data points lie on or near a line: Linear function $y = mx + b$ or $f(x)=mx + b$.
• If data points are increasing more and more rapidly: Exponential function $y = b^{x}$ or $f(x)=b^{x},b > 1$.
• If data points are increasing with rate of increase slowing down: Logarithmic function $y=\log_{b}x,b > 1$.
• If data points are decreasing and then increasing: Quadratic function $y = ax^{2}+bx + c$ or $f(x)=ax^{2}+bx + c,a>0$.
• If data points are increasing and then decreasing: Quadratic function $y = ax^{2}+bx + c$ or $f(x)=ax^{2}+bx + c,a < 0$.