17.06 Cubic functions (Extended)
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.
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 using technology.
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 using technology.
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 using technology.
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 using technology.
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, use technology to:
Find the y-intercept.
Find the x-intercepts.
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 using technology.
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 using technology.
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.
