Types of Graphs (Cambridge International IGCSE Mathematics)

Types of Graphs

What graphs do I need to know?

• You need to be able to recognise the following lines:

  • Straight lines

    • y = mx + c
    • Such as y = 3x * + 2, y = 5x * - 1, ...
    • Two important ones are y = x and y = -x

  • Horizontal lines

    • y = c
    • Such as y = 4, y = -10, ...

  • Vertical lines

    • x = k
    • Such as x = 2, x = -1, ...

• You need to be able to recognise quadratic graphs

  • y = x²
  • y = -x²
  • y = ax² + bx + c

• You also need to be able to recognise reciprocal graphs

  • $y=\frac{1}{x}$
  • $y=\frac{a}{x}$

What does a quadratic graph look like?

• The equation of a quadratic graph is y = ax² + bx + c

• A quadratic graph has either a u-shape or an n-shape

  • This type of shape is called a parabola

• u-shapes are called positive quadratics

  • because the number in front of x² is positive

    • For example, y = 2x² + 3x + 4

• n-shapes are called negative quadratics

  • because the number in front of x² is negative

    • For example, y = -3x² + 2x + 4

• You can plot quadratic graphs using a table of values

What are the key features of a quadratic graph?

• The point where the graph turns is called the vertex

  • Positive quadratics have a minimum point

    • The bottom of the u-shape

  • Negative quadratics have a maximum point

    • The top of the n-shape

• Quadratic graphs always have a vertical line of symmetry down the middle

  • The equation of the vertical line of symmetry is x = k

    • k is the x-coordinate of the minimum or maximum point

• Quadratic graphs do not have to cross the x-axis

  • If they do, two x-intercepts are created, called roots

    • If the curve just touches the x-axis, only 1 root is created

  • Roots are symmetric about the vertical line of symmetry

• Quadratic graphs always have one y-intercept

What does a reciprocal graph look like?

• The equation of a reciprocal graph is $y=\frac{a}{x}$

  • You cannot substitute in x = 0 (division by zero is not allowed)

    • $x\neq0$

  • You should not include x = 0 in a table of values

• The shape of $y=\frac{a}{x}$ is shown below (for a positive value of a)

  • It has two two curved branches

    • The branches are L-shaped
    • The L is more rectangular when a is smaller

  • The branches never connect!

• A negative value of a reflects the shape in the y-axis

Worked Example In each of the cases below, state the letter of the graph that corresponds to the equation given.

(a) $y=x+5$

Answer:

• This is a straight-line graph, y = mx + c*

Final answer: Graph D

(b) $y=-x^{2}+3x+2$

Answer:

This is a quadratic graph, y = ax² + bx +* c *(a = -1, b = 3, c = 2)

The number in front of* x*² is negative so it has an n-shape

Final answer: Graph C

(c) $y=\frac{4}{x}$

Answer:

This is a reciprocal graph, $y=\frac{a}{x}$ It has two L-shaped branches

Final answer: Graph B