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Mathematics 1.3 Investigate relationships between tables, equations and graphs

Quadratic equations and graphing parabolas

Graphing parabolas by plotting points

If an accurate graph is required plot points by drawing up a table of values. To do this:

• Use the values on the horizontal axis for the first row.
• Substitute in these numbers into the equation to work out the other coordinate.
• Plot the points and join them together with a smooth curve.
• Draw a rounded turning point.

Example

Draw the graph of y = x(x – 4) from x = 0 to x = 4

First draw up a table for each of the x values and work out the y-value for each by substituting the number for x in the equation.

Plot these points and join with a smooth curve.

Round the turning point.

For more information see:
Graphing Quadratic Equations I
Graphing Parabolas
Practice with Parabolas

Graphing parabolas by using transformations

The basic parabola y = x² can be transformed in the following ways:

• y = x² + 2 moves y = x² up 2 units
• y = x² – 2 moves y = x² down 2 units
• y = (x – 2)² moves y = x² to the right 2 units
• y = (x + 2)² moves y = x² to the left 2 units
• y = - x² reflects y = x² in the x-axis (it’s upside down)
  
  For more information on transforming graphs see:
  Function transformations I
  Graphs of quadratic functions
  Quadratic functions (Java™ required)
  Slider graph (Java™ required)

Graphing parabolas by using intercepts

1. To find where the graph cuts the x-axis, put y = 0 in the equation.
2. To find the y-intercepts, put x = 0 in the equation.
3. The axis of symmetry is halfway between the x-intercepts. Write an equation x = ?, The ? is where the line cuts the x-axis.
4. The vertex has the same x-coordinate as the axis of symmetry.
5. To find the y-intercept of the vertex, put the x-coordinate into the equation and calculate y.
   
   For more information on using intercepts, see:
   Curved graphs
   Slider graph (Java™ required)
   Parabolas in vertex form (Java™ required)

Quadratic equations, with a coefficient of x² other than ±1

In the equation y = kx², the number k in front of the x² means that the shape of the parabola changes.

If the number is greater than one, e.g. y = 3x², the parabola will be steeper.

If the number is less than one, e.g. y = x², the parabola will be flatter.

For more information on quadratics with a coefficient of x² other than ±1, see:
The meaning of the leading coefficient

Graphing Quadratic Equations

Quadratic functions (Java™ required)

Slider graph (Java™ required)

Writing equations

For excellence, assessment may involve writing equation(s) from a graph to solve a problem in context. For writing quadratic equations, a combination of two different types of transformations only is expected, eg y = 2x² + 3, or y = (x–2)² + 1.

If you are given a graph and asked to find its equation you firstly need to look at the coordinates of the vertex. If the vertex is at point (1,2) then you know that the graph of y = ax² (equation 1) has been translated 1 unit horizontally and 2 units vertically.

Therefore the equation is:
(y-2) = a(x-1)² (equation 2).

All you need to do now is work out the value of ‘a’. Do that by choosing some other point on the parabola (not the vertex) and substitute its coordinates into equation 2. You can then work out the value of ‘a’ and complete your equation.

Graphing Quadratic Equations

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