2.01 Cubic functions
Is the function y = - 2 x^{3} - 4 one-to-one?
By considering the graph of y = x^{3}, determine the following:
As x approaches infinity, what happens to the corresponding y-values?
As x approaches negative infinity, what happens to the corresponding y-values?
Determine whether the graphed function shown has an even or odd power. Explain your answer.
Consider the graph of the function y = x^{3}:
For what values of x is the functions concave down?
For what values of x is the functions concave up?
At what point does the concavity of the curve change?
What is this point called?
Consider the graph of the function:
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.
How does the graph of y = \dfrac {1}{2} x^{3} differ to the graph of y = x^{3}?
Consider the graph of the cubic function shown. For which values of x is y \geq 0?
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
Consider the graph of the function with equation of the form 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.
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)
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.
A cubic function has the equation y = x \left(x - 4\right) \left(x - 3\right). How many x-intercepts does it have?
Consider the given graph of a cubic function:
Determine whether the cubic is positive or negative.
State the coordinates of the y-intercept.
Which of the following could be the equation of the function?
If the graph of y = x^{3} is moved to the right by 10 units, what is the new equation?
Consider the cubic function y = - x^{3}
Complete the following table of values.
| x | -2 | -1 | 0 | 1 | 2 |
|---|---|---|---|---|---|
| y |
Sketch the graph of the curve.
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.
For each of the functions below:
Is the cubic increasing or decreasing from left to right?
Is the cubic more or less steep than the cubic y = x^{3} ?
What are the coordinates of the point of inflection?
Sketch the graph.
y = 2 x^{3} + 2
y = - \dfrac {x^{3}}{4} + 2
y = - 2 \left(x - 2\right)^{3}
y = \dfrac {1}{2} \left(x - 3\right)^{3}
y = - \dfrac {1}{2} \left(x - 1\right)^{3} + 2
Consider the curve y = - 3 \left(x - 1\right)^{3} + 3.
Find the x-intercept.
Find the y-intercept.
State the point of inflection.
Sketch the graph of the curve.
Consider the graph of y = x^{3}.
How do we shift the graph of y = x^{3} to get the graph of y = \left(x - 2\right)^{3} - 3?
Hence plot y = \left(x - 2\right)^{3} - 3.
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.
For each of the following curves:
Find the x-intercept(s).
Find the y-intercept(s).
Sketch the graph of the curve.
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)
Consider the equation y = 3 x + 2 x^{2} - x^{3}.
Express the equation in factorised form.
Find the y-intercept.
Find the x-intercepts.
Sketch the graph of the curve.