Modelling with Quadratic Functions

Key Concepts

• Define the vertical motion model.
• Assess the fit of a function by analyzing residuals.
• Explain to fit a quadratic function to data.

Vertex form of the quadratic function 

• The function f(x) = a(x−h)²+k , a≠0 is called the vertex form of a quadratic function 
• The vertex of the graph g is (h, k). 
• The graph of f(x) = a(x−h)²+k is a translation of the function f(x) = ax² that is translated h units horizontally and k units vertically. 

The standard form of the quadratic function 

• The standard form of a quadratic function is ax²+bx+c = 0 , a≠0
• The axis of symmetry of a standard form of quadratic function f(x) = ax²+bx+c is the line x = −b/2a. 
• The y-intercept of f(x) is c. 
• The x-coordinate of the graph of f(x) = ax²+bx+c is –b/2a. 
• The vertex of f(x) = ax²+bx+c is (–b/2a, f(–b/2a)). 

Vertical motion model 

When George throws the ball, it moves in a parabolic path. 

So, we can relate such real-life situations with quadratic functions. 

If the ball was hit with an initial velocity v₀, and h₀ be the initial height where the ball was hit. The height h (in feet) of the ball after some time (t seconds) can be calculated by the quadratic function:

h = -16t²+v₀t+h₀.  

This is called the vertical motion model. 

Assess the fit of the function by analyzing residuals 

A shopkeeper increases the cost of each item according to a function −8×2+95x+745−8×2+95x+745. Find how well the function fits the actual revenue. 

Step 1: Find the predicted value for each price increase using the function. 

For x=0, −8(0)²+95(0)+745 = 745

For x=1, −8(1)²+95(1)+745 = 832

For x=2, −8(2)²+95(2)+745 = 903

For x=3, −8(3)²+95(3)+745 = 958

For x=4, −8(4)²+95(4)+745 = 997

Subtract the predicted value from the actual revenues to find the residues.             

Residual = observed – predicted 

Step 2: Make a scatterplot of the data and graph the function on the same coordinate grid. 

Step 3: Make a residual plot to show the fit of the function of the data. 

Step 4: Assess the fit of the function using the residual plot.  

The residual plot shows both positive and negative residuals, which indicates a generally good model. 

Fit a quadratic function to data 

We know that to find the equation of the straight line that best fits a set of data, we use linear regression. 

Quadratic regression is a method used to find the equation of the parabola (quadratic function) that best fits data.  

Step 1: Using the graphing calculator, enter the values of x and y. 

Step 2: Use the quadratic regression feature. 

R-squared is the coefficient of the determination. 

The closes R² is to 1, the better the equation matches the given data points. 

Step 3: Graph the data and quadratic regression.  

Exercise

1. A rectangular wall has a length seven times the breadth. It also has a 4-ft wide brick border around it. Write a quadratic function to determine the area of the wall.

2. The data are modelled by f(x) = -2x²+16.3x+40.7. What does the graph of the residuals tell you about the fit of the model?

3. Write a function h to model the vertical motion, given h(t) = -16t²+v0t+h0. Find the maximum height if the initial vertical velocity is 32 ft/s, and the initial height is 75 ft.

4. Using a graphic calculator to find a quadratic regression for the data set.

What have we learned

• We can relate to real life situations using quadratic functions

Concept Map 

We can relate to real−life situations using quadratic functions. 

• To find the height of an object, we can use the vertical motion model.