## 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) = ax^{2}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 ax
^{2}+bx+c = 0 , a≠0 - The axis of symmetry of a standard form of quadratic function f(x) = ax
^{2}+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) = ax
^{2}+bx+c is –b/2a. - The vertex of f(x) = ax
^{2}+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 v_{0}, and h_{0 }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 *= -16t^{2}+v_{0}t+h_{0}.

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 R^{2} 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) = -2x^{2}+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) = -16t^{2}+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.

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