Cubic function graph

Here you will learn about cubic function graphs, including how to recognize them, how to sketch them and how to use them to estimate solutions.

Students will first learn about cubic function graphs as part of functions in high school.

What is a cubic function graph?

A cubic function graph is a graphical representation of a cubic function.

A cubic is a polynomial which has an x^3 term as the highest power of x.

These graphs have:

• a point of inflection where the curvature of the graph changes between concave and convex

• either zero or two turning points (also referred to as critical points or local minimum/maximum)

A cubic graph with two turning points can touch or cross the x axis between one and three times.

The end behavior describes y as x approaches infinity or negative infinity. Cubic function graphs that are increasing have y values that increase as x increases.

For example,

In the graph below, the leading coefficient of the x^3 term is positive, so the graph increases.

Also notice, the y -intercept of the curve is + \, 2 and the equation ends with + \, 2 for when x=0.

Cubic function graphs that are decreasing have y values that decrease as x increases.

For example,

In the graph below, the leading coefficient of the x^3 term is negative, so the graph decreases.

Also notice, the y -intercept of the curve is - \, 2 and the equation ends with - \, 2 for when x=0.

What is a cubic function graph?

Common Core State Standards

How does this relate to high school math?

• Interpreting Functions (HS.F.IF.C.7)
  Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

How to identify a cubic graph

In order to match a cubic function graph to its equation:

1. Identify any linear, quadratic or other type of function.
2. Identify the coefficient of the \textbf{x}\bf{^3} term as positive or negative.
3. Identify your final answer.

Identifying a cubic graph examples

Example 1: recognizing cubic graphs

Identify the correct graph for the equation:

1. Identify any linear, quadratic or other type of function.

Graph A is a straight line – it is a linear function.

Graph B is a parabola – it is a quadratic function.

Neither Graph A or Graph B is a cubic function.

2Identify the coefficient of the \textbf{x}\bf{^3} term as positive or negative.

Graph C and Graph D could be cubic functions. They have two turning points.

The given equation has a positive x^3 term.

For Graph C the graph decreases, so Graph C cannot be the correct graph.

Graph D is increasing.

3Identify your final answer.

Graph D is the correct graph for the equation.

Example 2: recognizing cubic graphs

Identify the correct graph for the equation:

How to sketch a cubic function graph from cubic equations

In order to plot a cubic function graph from cubic equations:

1. Complete the table of values.
2. Plot the coordinates.
3. Draw a smooth curve through the points.

Example 3: sketching a cubic function graph

Draw the curve of the equation for - \, 2\leq{x}\leq2.

Complete the table of values.

Plot the coordinates.

Draw an x -axis and y -axis or use a pre-made coordinate grid. Plot the coordinates:

(– \, 2, – \, 3); \, (– \, 1, 1); \, (0, – \, 1); \, (1, – \, 3); \, (2, 1).

Draw a smooth curve through the points.

The y -intercept of the curve is - \, 1 and appears to be the inflection point. Based on the other points, you can infer that the points (– \, 1, 1) and (1, – \, 3) are turning points.

To calculate the exact turning points, you can set the derivative equal to 0. But note that this strategy is not modeled on this page.

Example 4: sketching a cubic function graph

Draw the curve of the equation for - \, 3\leq{x}\leq3

Complete the table of values.

Plot the coordinates.

Draw an x -axis and y -axis or use a pre-made coordinate grid. Plot the coordinates:

(– \, 3, – \, 8); \, (– \, 2, 5); \, (– \, 1, 6); \, (0, 1); \, (1, – \, 4); \, (2, – \, 3); \, (3, 10).

Draw a smooth curve through the points.

The y -intercept of the curve is + \, 1 and appears to be the inflection point. Based on the other points, you can infer that the turning points are between (– \, 2, 5) and (– \, 1, 6) and (1, – \, 4) and (2, – \, 3).

To calculate the exact turning points, you can set the derivative equal to 0. But note that this strategy is not modeled on this page.

How to use a cubic function graph to estimate solutions

In order to use a cubic function graph to estimate solutions:

1. Find the given value on the \textbf{y} -axis.
2. Draw a straight horizontal line across the curve.
3. Draw a straight vertical line from the curve to the \textbf{x} -axis.
4. Read off the value on the \textbf{x} -axis.

Example 5: estimate the solution using a cubic function graph

Use the graph of y=x^3-5x+1 to find an approximate solution of the equation

Find the given value on the \textbf{y} -axis.

The value of 6 is in place of the y. Look on the y -axis for the value of 6.

Draw a straight horizontal line across the curve.

Use a digital tool or ruler to be as accurate as possible.

Draw a straight vertical line from the curve to the \textbf{x} -axis.

Example 6: estimate the solution using a cubic function graph

Use the graph of y=x^3-7x-3 to find approximate solutions of the equation

Find the given value on the \textbf{y} -axis.

The value of 2 is in place of the y. Look on the y -axis for the value of 2.

Draw a straight horizontal line across the curve.

Use a digital tool or ruler to be as accurate as possible. The horizontal line cuts the curve in 3 places.

Draw a straight vertical line from the curve to the \textbf{x} -axis.

There are 3 vertical lines required here.

Teaching tips for cubic function graph

• Before graphing cubic functions, students should be familiar with the coordinate plane and have experience graphing linear equations.

• Give students opportunities to compare cubic function graphs to graphs of other polynomial functions, asking them to share what they notice and wonder.

• Let struggling students use the step-by-step guides given above, until they feel comfortable with the material.

Easy mistakes to make

• Not paying attention to the scale on the \textbf{x} and \textbf{y} axis
  Cubic graphs often have different scales on the x -axis and the y -axis. Always pay close attention to this when you are plotting coordinates.

• Incorrectly calculating with exponents
  When using an equation to sketch a graph, remember that the exponent indicates multiplication. It tells you how many times to multiply the variable by itself. So, x^3=x \cdot x \cdot x and x^2=x \cdot x.

• Not drawing a smooth cubic curve
  The graph of the function needs to be drawn with a smooth curve. Avoid straight line segments for cubic function graphs. For example,
  
  

• Having more than \bf{2} turning points in the cubic function graph
  Make sure that your cubic graph has only one local minimum point and one local maximum point (if it has 2 turning points). If it has more, check that you have the correct values and that you have plotted them correctly.
  
  

• Confusing a cubic equation with a quadratic equation
  Some cubic equations include a term with x^2, but that does not make them quadratic equations.
  For example,
  
  Cubic equation: x^3+2 x^2+4 \hspace{1.5cm} Quadratic equation: 2 x^2+4
  
  

Related types of graphs lessons

• Reciprocal graph
• Exponential function graph
• Equation of a circle

The next lessons are

• Graphing linear equations
• Rate of change
• System of equations

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