Consider the cubic function y = - \dfrac{x^{3}}{4} + 2
Is the cubic increasing or decreasing from left to right?
Is the function more or less steep than function y = x^{3} ?
What are the coordinates of the point of inflection of the function?
Sketch the graph y = - \dfrac{x^{3}}{4} + 2
Consider the hyperbolic function y = \dfrac{3}{x} - 3
Which curve approaches positive and negative infinity more quickly, y = \dfrac{1}{x} or \\ y = \dfrac{3}{x} - 3?
What is the equation of the:
Vertical asymptote
Horizontal asymptote
Sketch the graph of y = \dfrac{3}{x} - 3.
Consider the hyperbolic function y = \dfrac{3}{x - 1} - 2
What is the equation of the:
Vertical asymptote
Horizontal asymptote
Sketch the graph y = \dfrac{3}{x - 1} - 2
Determine the equation of the new curve after performing the following transformations:
The curve y = x^{3} is reflected across the x-axis and then translated 3 units up.
The curve y = x^{3} is translated 2 units down and then reflected across the x-axis.
A graph of y = x^{2} is shown here. Sketch the curve after it has undergone transformations resulting in the function y = 4 x^{2} - 2.
For each of the following functions:
State whether the cubic is increasing or decreasing from left to right.
State whether the function is more or less steep than the function y = x^{3} .
State the coordinates of the point of inflection.
Sketch the graph of the functions.
For each of the following functions:
State the range of the function.
Sketch the graph the function.
Determine the coordinates of the vertex.
Consider the equation y=3\left|x+2\right|.
State the range of the function.
Sketch the graph the function.
State the coordinates of the vertex.
Consider the function y = 4^{x}. Find the equation of the new function that results from the following transformations:
The function is first reflected across the x-axis.
This new function is then multiplied by - 2.
State the pair of transformations on y = x^{3} that would result in y = - x^{3} - 3.
Consider the function y = 3^{ - x }-1.
Find the y-intercept of the curve y = 3^{ - x }-1.
Find the horizontal asymptote of the curve y = 3^{ - x }-1.
Sketch the graph of y = 3^{ - x }-1.
Write the equation when the graph of y = \log_{4} x is translated seven units downward, six units to the left, and then reflected in the x-axis.
Use the graph of y = \left|x\right| to graph \\ y = \left|x - 4\right| + 4.
Consider the graph of y = \log_{3} x.
What transformation must be done to obtain the graph of y = \log_{3} \left(x + 2\right) - 4 from y = \log_{3} x.
Hence sketch the graph of \\ y = \log_{3} \left(x + 2\right) - 4.
If the graph of y = \left|x\right| is stretched vertically by a factor of 4 and reflected across the x-axis, what is the equation of the new graph?
Consider the function f \left( x \right) = - \dfrac{5}{x}.
What transformation must be done to obtain the graph of f \left( x \right) from y = \dfrac{1}{x}.
Sketch the graph of f \left( x \right).
What is the domain of f \left( x \right)?
What is the range of f \left( x \right)?
Is f \left( x \right) increasing or decreasing over its domain?
A graph of y = x^{3} is shown:
A graph of y = x^{3} is shown:
What transformation must be done to obtain the graph of y = \left(x - 2\right)^{3} - 3 from y = x^{3}?
Hence Sketch the graph of y = \left(x - 2\right)^{3} - 3.
Consider the function y = 3 \times 2^{x} + 2.
Find the y-intercept of the curve.
Fill in the table of values for y = 3 \times 2^{x} + 2.
| x | - 3 | - 2 | - 1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| y |
Find the horizontal asymptote of the curve.
Sketch the graph of y = 3 \times 2^{x} + 2.
A graph of y = \log_{3} x is shown. Sketch the function after it has undergone transformations resulting in the equation y = - 2 \log_{3} \left(x - 2\right).
If the graph of y = x^{4} is moved to the right by 8 units and up by 6 units, what is its new equation?
If the graph of y = \left|x\right| is translated 5 units left and 3 units down, what is the equation of the new graph?
Consider the parabola y = 8 + 3 \left(x - 7\right)^{2}.
What is the vertex of the parabola?
The parabola is reflected across the y-axis. What will its new equation be?
What will be the vertex of the new parabola formed after the reflection?
Consider the functions F \left(x\right) = 3^{ - x } and G \left(x\right) = - 3^{x}. What are the two transformations that are required to turn the graph of F \left(x\right) into the graph of G \left(x\right)?
Consider the equation y = \left|4 - 2 x\right|.
State the range of the function.
Sketch the graph of the function.
Find the coordinates of the vertex.
Consider the graph of y = \sqrt{25 - x^{2}}:
What transformation must be done to obtain the graph of \\ y = \sqrt{25 - \left(x + 4\right)^{2}} - 2 from \\ y = \sqrt{25 - x^{2}}\text{?}
Hence sketch the graph of \\ y = \sqrt{25 - \left(x + 4\right)^{2}} - 2.
Consider the function y = 2 - 4^{ - x }.
Determine the y-intercept of its graph.
Is this an increasing or decreasing function?
As x approaches infinity, what function value does y approach?
Sketch the graph of y = 2 - 4^{ - x }.
Write down the equation of the new circle after x^{2} + y^{2} = 49 is translated:
5 units upwards
5 units downwards
5 units to the right
5 units to the left and 6 units upwards
For each of the following functions:
Find the x-intercept.
Complete the table of values for each function:
| x | \dfrac{1}{2} | 1 | 2 | 4 |
|---|---|---|---|---|
| y |
State the equation of the vertical asymptote.
Sketch the graph of the functions.
For the following functions:
Find the x-intercept.
Complete the table of values each function:
| x | - 9 | - 3 | - 1 | - \dfrac{1}{3} |
|---|---|---|---|---|
| y |
State the equation of the vertical asymptote.
Sketch the graph of the functions.
Write the equation corresponding to the graph of y = 7^{x} after having been translated 5 units upward, 3 units to the left then reflected in the x-axis.
Consider the function: y=-4\left|x-4\right|+3
State the domain of the function.
State the range of the function.
Sketch the graph of y=-4\left|x-4\right|+3.
Consider the function: y = - \dfrac{\left|x - 2\right|}{3} + 4.
State the domain of the function.
State the range of the function.
Sketch the graph of y = - \dfrac{\left|x - 2\right|}{3} + 4.
A graph of the function f \left( x \right) = 4 + \dfrac{4}{x - 5} is shown below:
Complete the following statements.
If x > 5, what value does the function approach as x approaches 5.
If x < 5, what value does the function approach as x approaches 5.
As x \to \infty, what does the function approach?
As x \to - \infty, what does the function approach?
State the equation of the vertical asymptote of f \left( x \right).
State the equation of the horizontal asymptote of f \left( x \right).
A parabola of the form y = \left(x - h\right)^{2} + k is symmetrical across the line x = 2, and its vertex lies 6 units below the x-axis.
Find the equation of the parabola.
Sketch the graph of the parabola.
Determine if the following parabolas have no x-intercepts:
y = \left(x - 7\right)^{2} + 4
y = - \left(x - 7\right)^{2} + 4
y = - \left(x - 7\right)^{2} - 4
y = \left(x - 7\right)^{2} - 4
The graph of y = \log_{4} x has a vertical asymptote at x = 0. By considering the transformations that have taken place, state the equation of the vertical asymptote of the following functions:
y = 2 \log_{4} x - 4
y = 2 \log_{4} x
y = \log_{4} \left(x - 5\right)
y = - \log_{4} x
y = \log_{4} \left(x + 3\right) - 2
Consider the function that has been graphed:
Determine the equation of the graph for x \geq 2.
Determine the equation of the graph for x < 2.
What transformation was applied to the graph of y = \left|x\right| to obtain the given graph?
Hence or otherwise, state the equation of the graph for all real x.
The exponential function y = 5^x has been transformed to become y = 5^{ 5 x + 4} - 2.
State the horizontal translation performed.
State the vertical translation performed.
State the dilation factor used.
Consider the function that has been graphed:
What transformation is applied to the graph of y = \left|x\right| to obtain the given graph?
Determine the equation of the graph for x \geq 0.
Determine the equation of the graph for x < 0.
Hence or otherwise, state the equation of the graph for all real x.
Consider the function f \left( x \right) = \left( - x \right)^{3} - 3. What transformations must be done to the graph of y = x^{3} to get the graph of f \left( x \right)?
If the graph of y = \left|x\right| is compressed vertically by a factor of \dfrac{1}{4}, reflected across the x-axis and translated 3 units left and 3 units up, what is the equation of the new graph?
Consider the following graph:
What transformations are applied to the graph of y = x^{3} to obtain the plotted graph?
Write down the equation of the curve.
The function: y = - \left(x + 5\right)^{2} - 1 is generated after performing the following transformations on a certain graph:
- Reflection across the x-axis
- Translation 2 units to the left
- Translation 5 units down
What is the equation of the original graph?
For the function y = - 10 \left(x + 8\right)^{2} - 9, state the transformations that have occurred if the initial function was y = x^{2}.
The functions f \left( x \right) and g \left( x \right) = f \left( k x\right) have been graphed:
What transformation can you notice in the graph?
Determine the value of k.
Use the graph of y = f \left( x \right) to sketch the graph of y = f \left( x \right) + 4.
Three functions have been graphed on the number plane. g \left(x\right) and h \left(x\right) are both transformations of f \left(x\right).
State the equation of f \left(x\right).
What transformation of f \left(x\right) can be used to describe g \left(x\right)?
State the equation of g \left(x\right).
What transformation of f \left(x\right) can be used to describe h \left(x\right)?
State the equation of h \left(x\right).
A graph of y = f \left( x \right) is shown:
Use the graph of f \left( x \right) to graph the function g \left( x \right) = f \left( - x \right).
Suppose that f is a function, and that \left(6, - 7 \right) is a point on the graph of y = f \left( x \right). If the function g is given by g \left( x \right) = f \left( x - 5 \right), find the corresponding point on the graph of \\ y = g \left( x \right).
Suppose that f is a function, and that \left(9, - 12 \right) is a point on the graph of y = f \left( x \right).
If the function g is given by g \left( x \right) = 6 f \left( x \right), find the corresponding point on the graph of \\ y = g \left( x \right).
The graph of y = P \left(x\right) is shown. Sketch the graph of y = 2 P \left(x\right).
Consider the graph of y=f\left(x\right):
Sketch the graph of g(x)= 2f(-x).
Sketch the graph of g(x) = 0.5f(2x).
Consider the graph of y=f(x):
Sketch the graph of g(x) = 2f(x-3).
Sketch the graph of g(x) = -f(0.5x).
Use the graph of y = f \left(x\right) to sketch the graph of y = \dfrac{1}{2} f \left(x + 4\right) - 2.
Use the graph of y = f \left(x\right) to sketch the graph of y = - \dfrac{1}{2} f \left(x - 6\right) - 2.
If the maximum value of y = f \left(x\right) is 6, what is the maximum value of y = \dfrac{f \left(x + 2\right)}{2} - 3?
Suppose that \left( - 4 , 3\right) is a point on the graph of y = g \left( x \right). Find the corresponding point on the graph of:
y = g \left( x + 7 \right) - 6
y = - 6 g \left( x - 4 \right) + 6
y = g \left( 6 x + 1 \right)