Do the following graphed functions have an even or odd power?
Consider the function y = x^{2}.
Complete the following table of values:
| x | - 3 | - 2 | - 1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| y |
Using the points in the table, plot the curve on a cartesian plane.
Are the y-values ever negative?
Write down the equation of the axis of symmetry.
What is the minimum y-value?
For every y-value greater than 0, how many corresponding x values are there?
Consider the function f \left( x \right) = - x^{2}.
Does the graph rise or fall to the right?
Does the graph rise or fall to the left?
Consider the functions f(x) = - x^{4} and g(x) = - x^{6}.
Graph f(x) = - x^{4} and g(x) = - x^{6} on the same set of axes.
Which of the above functions has the narrowest graph?
Consider the functions f(x) = x^{3} and g(x) = x^{5}.
Graph f(x) = x^{3} and g(x) = x^{5}.
How would the graph of y = x^{7} differ to the graph of f(x) = x^{3} and g(x) = x^{5} ?
Consider the function y = x^{7}.
As x approaches infinity, what happens to the corresponding y-values?
As x approaches negative infinity, what happens to the corresponding y-values?
Sketch the general shape of y = x^{7}.
Sketch the general shape of y = - x^{7}.
Sketch the graph of the function f \left( x \right) = x^{5} - x^{3}.
Consider the function y = x^{4} - x^{2}.
Determine the leading coefficient of the polynomial function.
Is the degree of the polynomial odd or even?
Does y = x^{4} - x^{2} rise or fall to the left?
Does y = x^{4} - x^{2} rise or fall to the right?
Sketch the graph of y = x^{4} - x^{2}.
Consider the function which has intercepts \left( - 4 , 0\right), \left(2, 0\right) and \left(0, 3\right).
What is the lowest degree of a polynomial that goes through these points?
Sketch the graph of the quadratic function that has the given intercepts.
Consider the function f \left( x \right) = - 6 x^{2} - 4 x + 5 which is concave down.
State the coordinates of the y-intercept of the function.
How many x-intercepts does the function have?
Match each function to its correct graph:
The graph of y = P \left(x\right) is shown. Sketch the graph of y = - P \left(x\right).
Consider the function y = - x^{5} + x^{3}.
What does y approach as x \to -\infty?
What does y approach as x \to \infty?
What are the x-intercepts of f \left( x \right)?
What is the y-intercept of the function?
Complete the following table of values:
| x | - 2 | - 1 | 0 | 1 | 2 |
|---|---|---|---|---|---|
| y |
Sketch the graph of the function.
Consider the function f \left( x \right) = x^{7} - 9 x^{3} - 2.
What is the maximum number of real zeros that the function can have?
What is the maximum number of x-intercepts that the graph of the function can have?
What is the maximum number of turning points that the graph of the function can have?
Consider the function f \left( x \right) = - 4 x^{5} - 6 x^{2} - 2 x - 9.
What is the maximum number of real zeros that the function can have?
What is the maximum number of x-intercepts that the graph of the function can have?
What is the maximum number of turning points that the graph of the function can have?
Sketch the graph of the function f \left( x \right) = x \left(x + 3\right) \left(x - 3\right).
For each of the following functions:
Find the x-intercepts.
Find the y-intercept.
Sketch the graph.
Consider the function y = - \left(x - 1\right)^{2} \left(x^{2} - 9\right).
Find the x-intercepts.
Find the y-intercept.
Does the graph have y-axis symmetry, origin symmetry, or neither?
Sketch the graph of the function.
Consider the function y = - x \left(x + 2\right) \left(x - 2\right) \left(x - 3\right).
What does y approach as x \to -\infty?
What does y approach as x \to \infty?
What are the x-intercepts?
What is the y-intercept?
Complete the following table of values:
| x | - 2 | - 1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|
| y |
Sketch the graph of the function.
Consider the function y = x^{3} - 6 x^{2} + 3 x + 10.
Does the graph rise or fall to the right?
Does the graph rise or fall to the left?
Express the equation in factorised form.
Find the x-intercepts.
Find the y-intercept.
Sketch the graph of the function.
Consider the function y = x^{4} - 4 x^{2}.
State the leading coefficient.
Does the function rise or fall to the left?
Does the function rise or fall to the right?
Express the equation in factorised form.
State the x-intercepts.
Find the y-intercept.
Sketch the graph.
Consider the function y = - 4 x^{3} + 11 x^{2} - 5 x - 2.
What does y approach as x \to -\infty?
What does y approach as x \to \infty?
What are the possible integer or rational roots?
Complete a table of values to test for the roots of the polynomial.
What is the y-intercept of the function?
Sketch the graph.
Consider the function y = 2 x^{4} + 3 x^{3} - 2 x^{2} - 3 x.
What does y approach as x \to -\infty?
What does y approach as x \to \infty?
The polynomial has a linear factor of x. Write the polynomial as a product of x and a cubic polynomial.
Hence, write down one of the roots of the polynomial.
Hence, find the rational and integer roots of the cubic factor.
Sketch the graph of y = 2 x^{4} + 3 x^{3} - 2 x^{2} - 3 x.
Solve the following equations:
Consider the function f \left( x \right) = x^{4} - 7 x^{3} + 12 x^{2} + 4 x - 16.
Is 5 a zero of the function?
Is 4 a zero of the function?
Is - 1 a zero of the function?
Consider the function f \left( x \right) = \left(x + 1\right)^{2} \left(x - 6\right) \left(x + 2\right)^{5}.
What are the zeros of the function?
State the multiplicity of each zero by filling in the table:
| Zero | Multiplicity |
|---|---|
| - 1 | |
| 6 | |
| - 2 |
Consider the function f \left( x \right) = - 3 \left(x - 6\right)^{4} \left(x + 2\right)^{5} x^{3}.
What are the zeros of the function?
State the multiplicity of each zero by filling in the table:
| Zero | Multiplicity |
|---|---|
| 6 | |
| - 2 | |
| 0 |
The polynomial P \left( x \right) = x^{2} + k x + 8 has a zero at x = 4.
Find the other zero of P \left( x \right).
Find the value of k.
Consider the polynomial P \left( x \right) = x^{7} - 7 x^{4} - 7 x^{3} + 12. Use the constant term to write down all the possible rational zeros.
Consider the polynomial f \left( x \right) = x^{4} + 2 x^{3} + 42 x^{2} + 12 x + 42.
What are the possible rational zeros of f \left( x \right)?
What are the actual rational zeros of this polynomial?
The polynomial P \left( x \right) = x^{3} - 12 x - 9 has a zero at x = - 3. Find the other zeros of P \left( x \right).
The polynomial P \left( x \right) = x^{4} - 23 x^{2} + 112 has zeros at x = 4 and x = - 4. Find the other zeros of P \left( x \right).
Consider the polynomial:
f \left( x \right) = x^{4} + 4 x^{3} - 9 x^{2} - 26 x - 30
What are the possible rational zeros of f \left( x \right)?
What are the actual rational zeros of f \left( x \right)?
Consider the polynomial P \left( x \right) = x^{3} + 4 x^{2} + x - 6
Write down all the possible rational zeros.
The graph of P \left( x \right) is shown.
State which of the possible zeros listed in the previous part are actual zeros of P \left( x \right).
Factorise P \left( x \right).
Consider the polynomial P \left( x \right) = x^{3} + 4 x^{2} + x - 6
Write down all the possible rational zeros.
The graph of P \left( x \right) is shown.
State which of the possible zeros listed in the previous part are actual zeros of P \left( x \right).
Factorise P \left( x \right).
The cubic equation x^{3} + k x^{2} + m x + 12 = 0 has a double root at x = 2.
Find the other root of the equation.
Find the value of k.
Find the value of m.
Sketch the graph of the function.