3.01 Identify characteristics of cubic functions
Consider the graph of y = x^{3}.
As x becomes larger in the positive direction (ie x approaches infinity), what happens to the corresponding y-values?
As x becomes larger in the negative direction (ie x approaches negative infinity), what happens to the corresponding y-values?
Consider the given graph of a cubic function.
Determine whether the cubic is positive or negative.
State the coordinates of the y-intercept.
State the equation of the function.
Consider the cubic function y = \dfrac{1}{2} x^{3} + x.
Determine whether the cubic is positive or negative.
Sketch the graph of y = \dfrac{1}{2} x^{3} + x.
State the coordinates of the x-intercept.
Consider the graph of the function y = x^{3}.
Determine the point where the curve changes from being concave down to being concave up.
This is called the point of inflection.
Consider the graph of the function y = - x^{3}.
Out of the points A, B and C:
At which point is the curve concave up?
Which is the point of inflection?
At which point is the curve concave down?
For each of the following functions:
For what values of x is the cubic concave up?
For what values of x is the cubic concave down?
State the coordinates of the point of inflection.
A cubic function is defined as y = \dfrac{1}{2} x^{3} + 4.
Find the x-intercept of the function.
Find the y-intercept of the function.
Consider the graph of the function:
The equation of the function can be written as y = a x^{3} + b x^{2} + c x + d.
Is the value of a is positive or negative?
State the coordinates of the y-intercept.
For which values of x is the graph concave up?
For which values of x is the graph concave down?
State the coordinates of the point of inflection.
Consider the graph of the cubic function shown.
For what values of x is y \geq 0?
Determine whether the following statements are true of the graphs of y = \dfrac{1}{2} x^{3} and y = x^{3}.
One is a reflection of the other about the y-axis.
y increases more rapidly on y = \dfrac{1}{2} x^{3} than on y = x^{3}.
y = \dfrac{1}{2} x^{3} is a horizontal shift of y = x^{3}.
y increases more slowly on y = \dfrac{1}{2} x^{3} than on y = x^{3}.
The graph of y = x^{3} has a point of inflection at \left(0, 0\right). By considering the transformations that have taken place, find the point of inflection of each cubic curve below:
y = \dfrac{2}{3} x^{3}
y = x^{3} + 3
y = - x^{3} + 4
For each cubic function below:
Complete a table of values of the form:
| x | -2 | -1 | 0 | 1 | 2 |
|---|---|---|---|---|---|
| y |
Sketch the graph.
Consider the curve y = x^{3} - 8.
Find the x-intercept.
Find the y-intercept.
Find the horizontal point of inflection.
Sketch the graph of the curve.
Consider the curve y = 3 x^{3} + 3.
Find the x-intercept.
Find the y-intercept.
State the coordinates of the point of inflection.
Sketch the graph of the curve.
Consider the equation y = x^{3} - 3.
Complete the following set of points for the given equation.
A(-3, ⬚), B(-2, ⬚), C(-1, ⬚), D(0, ⬚), E(1, ⬚), F(2, ⬚), G(3, ⬚)
Sketch the curve that results from the entire set of solutions for the equation being graphed.
Consider the cubic function y = x^{3} - 4.
State the y-intercept of the function.
Complete the following table of values.
| x | -1 | 1 | 2 |
|---|---|---|---|
| y |
Find the domain of the function in interval notation.
Find the range of the function in interval notation.
Sketch the graph.
Bianca completed the table of values for the equation y = 20 - x^{3}.
| x | -6 | -4 | -2 | 0 | 2 | 4 |
|---|---|---|---|---|---|---|
| y | 236 | 84 | 28 | 20 | -4 | -44 |
One of the points in the table is incorrect. Which point is it?
Bianca wants to find one other pair of values that satisfy y = 20 - x^{3} before graphing the curve. Find the ordered pair when the x-coordinate is 6.
Plot the complete set of solutions for y = 20 - x^{3}, making sure that the curve goes through all points that satisfy it.
Consider the curve y = - 2 x^{3} + 16.
Find the x-intercept.
Find the y-intercept.
State the coordinates of the point of inflection.
Find the domain of the function in interval notation.
Find the range of the function in interval notation.
Sketch the graph of the curve.
A graph of f(x) = x^{3} is shown. Sketch the curve after it has undergone transformations resulting in the function g(x) = x^{3} - 4.
A graph of y = x^{3} is shown. Sketch the curve after it has undergone transformations resulting in the function y = 2 \left(x - 2\right)^{3} - 2.
For the following cubic functions:
Determine whether the cubic is increasing or decreasing from left to right.
Determine whether the cubic is more or less steep than the cubic y = x^{3}.
Find the coordinates of the point of inflection of the cubic.
Sketch the graph.
y = 2 x^{3} + 2
y = 4 x^{3} - 3
y = - \dfrac{x^{3}}{4} + 2
Consider the graph of y = x^{3} shown:
How do we shift the graph of y = x^{3} to get the graph of y = \left(x - 2\right)^{3} - 3 ?
Hence, sketch y = \left(x - 2\right)^{3} - 3.
Consider the function y = 2 \left(x - 2\right)^{3} - 2
Is the cubic increasing or decreasing from left to right?
Is the function more or less steep than the function y = x^{3} ?
What are the coordinates of the inflection point of the function?
Sketch the graph y = 2 \left(x - 2\right)^{3} - 2.
Consider the curve y = 2 \left(x + 1\right)^{3} + 16.
Find the x-intercept.
Find the y-intercept.
State the coordinates of the point of inflection.
Sketch the graph of the curve.
Consider the curve y = - 3 \left(x - 1\right)^{3} + 3.
Find the x-intercept.
Solve for the y-intercept.
State the coordinates of the point of inflection.
Sketch the graph of the curve.
The cubic function y = \left(x + 1\right) \left(x - 2\right) \left(x - 3\right) has been graphed below. Determine the number of solutions to the equation \left(x + 1\right) \left(x - 2\right) \left(x - 3\right) = 0.
A cubic function has the equation y = x \left(x - 4\right) \left(x - 3\right). How many x-intercepts will it have?
State whether the following functions pass through the origin:
y = \left(x - 2\right)^{2} \left(x + 3\right)
y = \left(x + 1\right)^{3}
y = \left(x - 4\right) \left(x + 7\right) \left(x - 5\right)
y = x \left(x - 6\right) \left(x + 8\right)
State whether each function has exactly two x-intercepts:
y = \left(x + 3\right)^{3}
y = \left(x + 6\right)^{2} \left(x + 5\right)
y = \left(x + 7\right) \left(x - 1\right) \left(x - 4\right)
y = x \left(x - 2\right) \left(x - 8\right)
Consider the cubic function y = \left(x + 3\right) \left(x - 2\right) \left(x - 5\right).
Determine the x-intercepts.
A second cubic function has the same x-intercepts, but is a reflection of the above function about the x-axis. State the equation of the reflected function.
Sketch the function y = \left(x - 2\right) \left(x + 1\right) \left(x + 4\right) showing the general shape of the curve and the x-intercepts.
Sketch the graph of f(x) = \left(x - 2\right)^{2} \left(x - 4\right) and g(x) = - \left(x - 2\right)^{2} \left(x - 4\right) on the same number plane.
Consider the function y = \left(x + 3\right)^{3}.
Complete the following table of values:
| x | -5 | -4 | -3 | -2 | -1 |
|---|---|---|---|---|---|
| y |
Sketch the graph.
State the domain.
State the range.
Consider the function y = \left(x - 2\right)^{3}.
Complete the following table of values:
| x | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| y |
Sketch the graph.
Consider the function y = \left(x + 1\right)^{3}.
Complete the following table of values:
| x | -3 | -2 | -1 | 0 | 1 |
|---|---|---|---|---|---|
| y |
Sketch the graph.
Consider the equation y = \left(x - 4\right)^{3}.
Complete the set of solutions for the above equation.
A (3,⬚) , B (2,⬚) , C(5,⬚) , D (4,⬚) , E (6,⬚)
Plot the points on a coordinate axes.
On the same axes, plot the curve that results from the entire set of solutions for the equation being graphed.
Consider x = 6.17. According to the points on the graph, between which two integer values should the corresponding y-value lie?
For each of the functions below:
Find the x-intercepts.
Find the y-intercept.
Sketch the graph.
y = \left(x + 3\right) \left(x + 2\right) \left(x - 2\right)
y = - \left(x + 4\right) \left(x + 2\right) \left(x - 1\right)
y = \left(x - 2\right)^{2} \left(x + 5\right)
For each of the functions below:
Express the equation in factorised form.
Find the y-intercept of the graph.
Find the x-intercepts of the graph.
Sketch the graph of the curve.
y = 3 x + 2 x^{2} - x^{3}
y = x^{3} - 4 x^{2} - 7 x + 10
The volume of a sphere has the formula V = \dfrac{4}{3} \pi r^{3}. The graph relating r and V is shown:
Fill in the following table of values for the equation V = \dfrac{4}{3} \pi r^{3}, in terms of \pi:
| r | 1 | 2 | 4 | 5 | 7 |
|---|---|---|---|---|---|
| V |
Determine whether the following intervals show the volume, V of a sphere that has a radius measuring 4.5\text{ m}.
Using the graph, what is the radius of a sphere of volume 288 \pi \text{ m}^{3} ?
A cube has side length 7\text{ cm} and a mass of 1715\text{ g}. The mass of the cube is directly proportional to the cube of its side length.
Let k be the constant of proportionality for the relationship between the side length x and the mass m of a cube. Find the value of k.
Hence state the equation relating the mass (m) and side length (x) of a cube.
Complete the table of values:
| x | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| m |
Sketch the graph of your equation.
From the equation, find the mass of a cube with side 8.5\text{ cm}, to the nearest gram.
A cube has a mass of 1920\text{ g}. From your graph, determine what whole number value its side length is closest to.
A box without a top cover is to be constructed from a rectangular cardboard, measuring 6\text{ cm} by 10\text{ cm} by cutting out four square corners of length x\text{ cm}. Let V represent the volume of the box.
Express the volume V of the box in terms of x, writing the equation in factorised form.
For what range of values of x is the volume function defined?
Sketch the graph of the volume function.
Determine the volume of a box that has a height equivalent to the shorter dimension of the base.
A cylinder of radius x and height 2 h is inscribed in a sphere of radius \sqrt{15}, centre at O as shown:
Form an equation relating x and h.
Form an expression for V, the volume of the cylinder, in terms of h.
Determine the domain of h.
The cylinder with the largest possible volume has a height of 2 \sqrt{5}, so h = \sqrt{5}.
Determine the exact volume of this cylinder.
