6.3 SUPPORTING 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

Linear graph (straight line) 

Quadratic graph , parabola

Quadratic graph = Parabola (smooth curve)

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

cup parabola

Cup parabola 

cap parabola

Cap parabola

cup
cap

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 it is a cup parabola or a cap parabola.

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

An easy way to remember is to draw a smiley.

happy face

Positive x2 is a happy smiley so, a cup parabola 

sad face

negative x2 is 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. 

one

Step 1: 

Make a table of y = x2 + 3 

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

table quadratic formulas
two

Step 2: 

Complete the table for y = x2 + 3 

Example calculation:  y = (-1)2 + 3 

                                             y = 1 + 3 = 4 

table quadratic formulas

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, parabola, cup

Plot the points.

cup parabola

Draw a smooth curve through the points.

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

Linear formulas versus Quadratic formulas

There are several differences between linear- and quadratic formulas. An example will be shown in the two columns underneath  

Formula

Linear formulas

The formula does not have a squared variable (letters) 

y = x + 1 

Quadratic formulas

The formula does have a squared variable (letters)

y = x2 + 1 

Table

Graph

Table with two points

Name of the graph : Line

Table with 7 points 

Name of the graph : Parabola

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) How about a width of  3 and -3 meters? 

d) Make a table for the formula h = -0.5w2 + 6.5 

e) What do you notice about the table? 

f) Draw the graph of h = -0.5w2 + 6.5