6.3 NORMAL PATH graphs

Drawing parabolas

Substituting numbers in quadratic formulas

Theory

The same as with a linear formula, you can draw a graph of a quadratic formula. Whereas the graph of a linear formula is a straight line, the graph of a quadratic formula is a smooth curve. This curve is called a parabola.

Linear graph (straight line)

Quadratic graph = Parabola (smooth curve)

There are two different types of parabolas: a cup parabola and a cap parabola.

Cup parabola

Cap parabola

Can you see the similarities between the pictures of the cup & the cap and the pictures of the cup parabola and the cap parabola?

The formula of the cup parabola above is: y = x² + 1

The formula of the cap parabola above is: y = -4x²- 2

Without drawing you could tell if the graph is going to be a cup parabola or a cap parabola.

Cup parabolas always have a positive x²and cap parabolas always have a negative x²

An easy way to remember is to draw a smiley.

Positive x²is a happy smiley so, a cup parabola.

Negative x²is a sad smiley so, a cap parabola.

Examples

How to draw a parabola?

Given the formula y = x²+ 3

Draw the graph of this formula.

Step 1:

Make a table of y = x²+ 3

Take for x the numbers -3 up and including 3.

Step 2:

Complete the table for y = x²+ 3

Example calculation: y = (-1)²+ 3

y = 1 + 3 = 4

Step 3:

Draw the graph for y = x²+ 3

* First draw a set of axis -> WITH a protractor triangle .

* Secondly, plot the points.

* Draw a smooth curve -> WITHOUT a protractor triangle

Plot the points.

Draw a smooth curve through the points.

Don't forget to label the x-axis and the y-axis!

Linear formulas versus Quadratic formulas

Linear formulas are formulas without a squared variable (= letter). If you are going to draw the graph of a linear formula, you are first going to make a table with at least two points. If you know two points you can plot the points and draw the graph. The graph is a straight line and therefore the graph is called a line. You must use a protractor triangle when drawing a linear graph.

Quadratic formulas are formulas with a squared variable (=letter). If you are going to draw the graph of a quadratic formula, you are first going to make a table with seven points. If you know the points you need to plot them and you need to draw the graph. The graph is a smooth curved graph called a parabola. Because of the smooth curve you may not use your protractor triangle.

How to check whether a point is on the graph or not?

To check whether a point is on the graph or not you need to fill the given point into your formula. Underneath you will find a couple of examples:

Given the formula y = x²- 3. Check with a calculation whether y = 22 when x = 5?

First: fill in x = 5

y = 5²- 3

y = 25 - 3

y = 22

Conclusion: So, y = 22 when x = 5

Given the formula y = 2x²+ 4. Check with a calculation if point A(3, 24) is on the graph of y = 2x²+ 4?

First: Fill in the given point. You know that x = 3 and y = 24.

24 = 2 ^.3² + 4

24 = 2 ^.9 + 4

24 = 18 + 4

24 = 22

This is NOT true.

Conclusion: So, point A(3, 24) is not on the graph of y = 2x²+ 4

Example with linear formulas

Exercises

Two students of Cambreur College are going to bungee jump. They are going to jump of a bridge. Before they are going to jump they want to know the height of the bridge. The instructor gives them the following formula to calculate the height.

h = -0.5w² + 6.5

Where h is the height in meters and w the width in meters.

a) What type of parabola is this?

b) What is the height when the students are going to jump at a width of 0 meters?

c) Draw the graph of h = -0.5w² + 6.5

Answers