Properties of Graphs (DP IB Applications & Interpretation (AI)): Revision Note

Quadratic Functions & Graphs

What are the key features of quadratic graphs?

• A quadratic graph is of the form where .

• The value of a affects the shape of the curve

  • If a is positive the shape is 

  • If a is negative the shape is 

• The y-intercept is at the point (0, c)

• The zeros or roots are the solutions to 

  • These can be found using your GDC or the quadratic formula

  • These are also called the x-intercepts

  • There can be 0, 1 or 2 x-intercepts

• There is an axis of symmetry at 

  • This is given in your formula booklet

  • If there are two x-intercepts then the axis of symmetry goes through the midpoint of them

• The vertex lies on the axis of symmetry

  • The x-coordinate is 

  • The y-coordinate can be found using the GDC or by calculating y when 

  • If a is positive then the vertex is the minimum point

  • If a is negative then the vertex is the maximum point

Cubic Functions & Graphs

What are the key features of cubic graphs?

• A cubic graph is of the form  where .

• The value of a affects the shape of the curve

  • If a is positive the graph goes from bottom left to top right

  • If a is negative the graph goes from top left to bottom right

• The y-intercept is at the point (0, d)

• The zeros or roots are the solutions to 

  • These can be found using your GDC

  • These are also called the x-intercepts

  • There can be 1, 2 or 3 x-intercepts

    • There is always at least 1

• There are either 0 or 2 local minimums/maximums

  • If there are 0 then the curve is monotonic (always increasing or always decreasing)

  • If there are 2 then one is a local minimum and one is a local maximum

Exponential Functions & Graphs

What are the key features of exponential graphs?

• An exponential graph is of the form

  • or where 

  • 

    • Where e is the mathematical constant 2.718…

• The y-intercept is at the point (0, k + c)

• There is a horizontal asymptote at y = c

• The value of k determines whether the graph is above or below the asymptote

  • If k is positive the graph is  above the asymptote

    • So the range is 

  • If k is negative the graph is below the asymptote

    • So the range is 

• The coefficient of x and the constant k determine whether the graph is increasing or decreasing

  • If the coefficient of x and k have the same sign then graph is increasing

  • If the coefficient of x and k have different signs then the graph is decreasing

• There is at most 1 root

  • It can be found using your GDC

Sinusoidal Functions & Graphs

What are the key features of sinusoidal graphs?

• A sinusoidal graph is of the form

  • 

  • 

• The y-intercept is at the point

  • (0, d) for 

  • (0, a + d) for 

• The period of the graph is the length of the interval of a full cycle

  • This is 

• The maximum value is y = a + d

• The minimum value is y = -a + d

• The principal axis is the horizontal line halfway between the maximum and minimum values

  • This is y = d

• The amplitude is the vertical distance from the principal axis to the maximum value

  • This is a