4.09 Sketching polynomials
For each of the following graphs of the function y = f \left( x \right):
Write the x-values of the x-intercepts.
Write the y-value of the y-intercept.
Determine the behaviour of the function as x \to \infty.
Determine the behaviour of the function as x \to -\infty.
Consider the function f \left( x \right) = x^{4}.
Complete the following table of values:
| x | -4 | -2 | -1 | 0 | 1 | 2 | 4 |
|---|---|---|---|---|---|---|---|
| f(x) |
Hence, sketch the curve.
Consider the cubic function f \left( x \right) = 2 x^{3} - 7.
Complete the following table of values:
| x | -5 | -1 | 0 | 1 | 5 |
|---|---|---|---|---|---|
| f(x) |
The function appears to be asymmetric about which point?
At which x-value would the function have a value between - 9 and - 257?
Does the function increase or decrease for values of x > 0?
Hence, sketch the curve.
Consider the cubic function f \left( x \right) = 4 x^{3} + 8 x^{2}.
Complete the following table of values:
| x | -8 | -3 | -1 | 0 | 1 | 7 |
|---|---|---|---|---|---|---|
| f(x) |
Hence, sketch the curve y = f \left( x \right).
State the x-values of the x-intercepts.
Find the function value at x = 2.
Does the function increase or decrease for values of x > 0?
Consider P \left( x \right) = x^{3} + 2 x^{2} - 5 x - 6 shown:
State all the possible rational zeros.
Determine the actual zeros of P \left( x \right).
Hence, factorise P \left( x \right) .
Consider P \left( x \right) = x^{3} - 3 x^{2} - 6 x + 8 shown:
State all the possible rational zeros.
Determine the actual zeros of P \left( x \right).
Hence, factorise P \left( x \right) .
Consider P \left( x \right) = x^{3} - 5 x^{2} - 29 x + 105 shown:
State all the possible integer zeros.
Determine the actual zeros of P \left( x \right).
Hence, factorise P \left( x \right) .
Consider P \left( x \right) = x^{3} - 4 x^{2} + x + 6 shown:
State all the possible rational zeros.
Determine the actual zeros of P \left( x \right).
Hence, factorise P \left( x \right) .
A polynomial has been graphed with each of its intercepts shown:
Let k be a non-zero real number. Determine whether the following polynomials could describe the graph:
y = k \left(x + 6\right)^{2} \left(x + 2\right) \left(x - 4\right)
y = k \left(x - 6\right)^{2} \left(x - 2\right) \left(x + 4\right)
y = k \left(x - 6\right) \left(x - 2\right) \left(x + 4\right)
y = k \left(x + 6\right) \left(x + 2\right) \left(x - 4\right)
Determine the value of k.
Hence, state the equation of the least degree polynomial for the displayed graph.
A polynomial has been graphed with each of its intercepts shown:
Let k be a non-zero real number. Determine whether the following polynomials could describe the graph:
y = k \left(x - 1\right)^{2} \left(x + 2\right)
y = k \left(x + 1\right)^{2} \left(x - 2\right)
y = k \left(x + 1\right)^{3} \left(x - 2\right)
y = k \left(x - 1\right)^{3} \left(x + 2\right)
Determine the value of k.
Hence, state the equation of the least degree polynomial for the displayed graph.
A polynomial has been graphed with each of its intercepts shown.
Let k be a non-zero real number. Determine whether the following polynomials could describe the graph:
y = k \left(x + 2\right)^{2} \left(x - 1\right) \left(x - 4\right)^{2}
y = k \left(x + 2\right) \left(x - 1\right)^{3} \left(x - 4\right)
y = k \left(x - 2\right)^{2} \left(x + 1\right) \left(x + 4\right)^{2}
y = k \left(x - 2\right)^{2} \left(x + 1\right)^{3} \left(x + 4\right)^{2}
Determine the value of k.
Hence, state the equation of the least degree polynomial for the displayed graph.
The volume of a sphere has the formula V = \dfrac{4}{3} \pi r^{3}. The graph relating r and V is shown below.
Complete the table of values, correct to two decimal places.
| r | 1 | 2 | 4 | 5 | 7 |
|---|---|---|---|---|---|
| V |
A sphere has radius measuring 3.5\text{ m}. Determine the exact interval that the volume of the sphere falls within.
According to the graph, what is the radius of a sphere of volume 288 \pi \text{ m}^3?
A cube has side length 5\text{ cm} and a mass of 375\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, such that m=kx^3. Solve for k.
Hence state the equation relating the mass and side length of the cube.
Sketch the curve of for this equation.
Find the mass of a cube with side 6.5\text{ cm}, to the nearest gram.
A cube has a mass of 576\text{ g}. Determine what whole number value its side length is closest to.
The mass in grams, M, of a cube of cork is given by the formula M = 0.28 l^{3}, where l is the side length of the cube in centimetres.
What is the mass of a cubic centimetre of cork to the nearest 0.01 of a gram?
Complete the table values, to the nearest 0.01 of a gram if necessary.
| \text{Side length(cm)} | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|
| \text{Mass (g)} |
Sketch the graph that represents the function.