Sketch the graph of y = 2^{x + 4}.
Consider the function y = - x^{5} - 5.
Sketch the general shape of the function y = - x^{5} - 5.
Sketch the graphs of y = - x^{5} and y = - x^{5} - 5 on the same number plane.
What is the y-intercept of the graph y = - x^{5} - 5?
Sketch the general shape of the curve y = x^{4}.
Consider the graph of y = 6^{ - x }:
What transformation must be used to obtain the graph y = - 6^{ - x } from y = 6^{ - x }.
Given the graph of y = 6^{ - x }, sketch the graph of y = - 6^{ - x }.
Describe the rate of change of the graph of y = - 6^{ - x }.
Consider the graph of y = 4^{x}:
What transformation must be done to obtain the graph of y = 4^{ - x } from y = 4^{x}.
Sketch the graph ofy = 4^{ - x }.
Consider the graph of y = 5^{x}:
What transformation must be done to obtain the graph of y = - 5^{x} from \\ y = 5^{x}?
Sketch the graph of y = - 5^{x}.
Consider the graph of y = f \left( x \right), describe the transformation required to get the graph of y = f \left( x \right) + 4.
Consider the function y = \left(x - 6\right)^{2}.
Complete the table of values:
What is the minimum value of y?
Hence state the coordinates of the vertex of the parabola.
How many units to the right has y = x^{2} been translated?
Translate the parabola y = x^{2} on the graph to sketch y = \left(x - 6\right)^{2}.
| x | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|
| y |
Consider the parabola y = x^{2} - 3.
Complete the table of values:
| x | -2 | -1 | 0 | 1 | 2 |
|---|---|---|---|---|---|
| y |
Use the graph of y = x^{2} to sketch the graph of y = x^{2} - 3.
What is the y-value of the y-intercept of the graph y = x^{2} - 3?
What type of transformation occurs on the graph when adding a constant to the equation y = x^{2}?
Consider the function y = x^{4} - 4.
Complete the table of values for y = x^{4}:
| x | -2 | -1 | 0 | 1 | 2 |
|---|---|---|---|---|---|
| y |
Use the graph of y = x^{4} to sketch the graph of y = x^{4} - 4.
What is the y-intercept of the graph \\ y = x^{4} - 4?
What type of transformation occurs on the graph when adding a constant to the equation y = x^{4}?
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 on a number plane.
What transformation of the graph y = x^{3} will result in the graph of y = \left(x - 2\right)^{3}?
A graph of y = x^{4} is shown. Sketch the curve after it has undergone transformations resulting in the function y = x^{4} - 2.
Consider the graph of y = \dfrac{1}{x}:
How we can trasform the graph of \\ y = \dfrac{1}{x} into the graph of y = \dfrac{1}{x + 2}?
Hence sketch the graph of y = \dfrac{1}{x + 2} on a number plane.
Consider the graph of f \left(x\right) = x.
Write down the equation of the new function g \left(x\right) which is formed by evaluating f \left(x\right) + 4.
Sketch the graph of y = g \left(x\right).
What transformation has the graph of f \left(x\right) undergone to become the graph of g \left(x\right).
Consider the graph of y = x^{2}:
What transformation must be done to obtain the graph of y = \left(x - 2\right)^{2} from y = x^{2}?
Hence sketch the graph of y = \left(x - 2\right)^{2} on a number plane.
Consider the graph of y = 3^{ - x }:
What transformation must be done to obtain the graph of y = 3^{ - x } + 2 from y = 3^{ - x }?
Hence sketch the graph of y = 3^{ - x } + 2 on a number plane.
Consider the graph of y = x^{3}:
Sketch the graph after it has undergone transformations resulting in the function y = x^{3} - 4.
Consider a graph of y = 3^{x}:
What type of transformation must be done to obtain the graph of y = 3^{x} - 4 from y = 3^{x}.
Sketch the graph of y = 3^{x} - 4 on a number plane.
Consider the graph of y = \dfrac{1}{x}:
What transformation must be done to obtain the graph of y = \dfrac{1}{x} + 3 from \\ y = \dfrac{1}{x}.
Hence sketch the graph of y = \dfrac{1}{x} + 3 on a number plane.
Consider the graph of y = \sqrt{4 - x^{2}}:
What transformation must be done to obtain the graph of y = \sqrt{4 - x^{2}} + 2 from the graph of y = \sqrt{4 - x^{2}}?
Hence sketch the graph of \\ y = \sqrt{4 - x^{2}} + 2 on a number plane.
Consider the graph of y = x^{4}.
Sketch the curve after it has undergone a transformation resulting in the function \\ y = \left(x-1\right)^{4}.
Consider the graph of y = - 5^{x}:
State the equation of the asymptote of y = - 5^{x}.
What would be the asymptote of \\ y = 2 - 5^{x}?
How many x-intercepts would \\ y = 2 - 5^{x} have?
If the graph of y = \left|x\right| is translated 5 units down, what is the equation of the new graph?
Consider the graphs of f \left(x\right) = \dfrac{3}{x} and g \left(x\right):
Write g \left(x\right) in terms of f(x).
State the equation of g \left(x\right).
Consider the functions f\left(x\right) = x^2-5 and g\left(x\right)=x^2-6. Write g(x) in terms of f(x).
Consider the function y = \dfrac{- 1}{x}.
What value cannot be substituted for x?
In which quadrants does y = \dfrac{- 1}{x} lie?
Consider y = \dfrac{- 1}{x - 4}. What value cannot be substituted for x?
In which quadrants does y = \dfrac{- 1}{x - 4} lie?
What transformation must be done to obtain the graph of y = \dfrac{- 1}{x - 4} from y = \dfrac{- 1}{x}?
Consider the graphs of f \left(x\right) = 3^{x} and g \left(x\right):
Write g \left(x\right) in terms of f(x).
State the equation of g \left(x\right).
Find the equation of the new cubic when the curve y = x^{3} + 2 is translated:
6 units up
6 units down
Consider the hyperbolic function y = \dfrac{- 2}{x - 1}
State whether the following indicates the position of the hyperbola's branches relative to its asymptotes?
What are the equations of the vertical and horizontal asymptotes?
Sketch the graph of y = \dfrac{- 2}{x - 1}.
Consider the function y = \left|x\right| + 2.
What is the domain of the function?
What is the range of the function?
Hence, sketch the graph of y = \left|x\right| + 2.
Describe how to transform the graph of y = g \left(x\right) to become the graph of y = g \left(x + 6\right).
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.
Hence or otherwise, state the equation of the graph for all real x.
What transformation must be done to obtain this graph from y = \left|x\right|?
Consider the graph of y = \sqrt{4 - x^{2}}:
State the new equation if the graph was moved downwards by 7 units?
State the new equation if the graph was moved to the left by 3 units?
State the equation of the function obtained by translating the graph of y = x^{3} to the right by 10 units.
The graph of y = 4 x has been shifted up 2 units. What is its new equation?
What transformation must be done to obtain the graph of y = a^{\left(x + 5\right)} from y = a^{x}?
Consider the graph of the hyperbola y = \dfrac{1}{x}:
What would be the new equation if the graph was shifted upwards by 4 units?
What would be the new equation if the graph was shifted to the right by 7 units?
Consider the function y = \log_{3} \left(x + 4\right).
As x increases, is the function increasing or decreasing?
Is the function more steep, less steep or equally as steep as y = \log_{3} x?
What is the equation of the vertical asymptote of y = \log_{3} \left(x + 4\right)?
Sketch the graph of y = \log_{3} \left(x + 4\right).
Consider the function y = \left|x - 2\right|.
State the domain of the function.
State the range of the function.
Sketch the graph of y = \left|x - 2\right|.
Consider the graph of y = x^{2} - 3:
What would be the new equation if the graph of y = x^{2} - 3 moved upwards by 5 units?
What would be the new equation if the graph of y = x^{2} - 3 moved to the right by 3 units?
What is the dilation factor of:
A parabola of the form y = a x^{2} goes through the point \left(2, - 8 \right).
Find the value of a.
Find the coordinates of the vertex.
Sketch the graph of the parabola.
The graph of y = x^{4} has been provided on the coordinate axes below. Sketch the graph of y = \dfrac{1}{2} x^{4}.
Consider the original graph y = 3^{x}. The function values of the graph are multiplied by 2 to form a new graph.
For each point on the original graph, find the point on the new graph:
| Point on original graph | Point on new graph |
|---|---|
| \left(-1,\dfrac{1}{3}\right) | \left(-1,⬚\right) |
| \left(0,1\right) | \left(0,⬚\right) |
| \left(1,3\right) | \left(1,⬚\right) |
| \left(2,9\right) | \left(2,⬚\right) |
What is the equation of the new graph?
Sketch the graphs of y = 3^x and the new graph on the same set of axes.
For negative x-values, is 2 \times 3^{x} above or below 3^{x}?
For positive x-values, is 2 \times 3^{x} above or below 3^{x}?
The graph of the function f\left(x\right), contains the points P\left(-2, 4\right) and Q\left(4, -5\right). Find two points on the graph of the following functions:
Sketch the graph of y = 3 \left|x\right|.
The function f \left(x\right) = x^{3} is transformed into the function g \left(x\right) = k x^{3}.
For each of the following values of k state the affect k will have on g(x):
k < - 1
- 1 < k < 0
0 < k < 1
k > 1
Sketch the graph of g \left(x\right) = - \dfrac{2}{5} x^{3}.
Consider the functions y = \dfrac{4}{x} and y = \dfrac{4}{2x}.
Which graph lies further away from the coordinate axes?
For hyperbolas of the form y = \dfrac{k}{x}, as k increases, describe what happens to the hyperbola.
Of the two functions y = 2^{x} and y = 3 \times 2^{x}, which is increasing more rapidly for x > 0?
Consider the functions y=2^{x} and y = 2^{3x}.
Sketch both functions on the same number plane.
Which function is increasing more rapidly for x\gt1?
Describe the affect on f \left(x\right) if it is transformed into a new function g \left(x\right) = f \left(\dfrac{x}{k}\right).
A function f\left(x\right) is transformed into a new function g\left(x\right) = f\left(\dfrac{x}{k}\right). If k\gt1, what effect will this have on the graph of g(x)?
The table below shows values that satisfy the function f \left(x\right) = \left|x\right|:
| x | - 3 | - 2 | - 1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| y = f \left(x\right) | 3 | 2 | 1 | 0 | 1 | 2 | 3 |
Complete the table of values for each transformation of the function f \left(x\right):
| x | - 3 | - 2 | - 1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| y = f \left(x\right) | 3 | 2 | 1 | 0 | 1 | 2 | 3 |
| g \left(x\right) = 5 f \left(x\right) | |||||||
| h \left(x\right) = - 2 f \left(x\right) |
Sketch the graphs of f \left(x\right) and g \left(x\right) on the same number plane.
Describe the transformation undergone by f \left(x\right) to become g \left(x\right).
Sketch the graphs of f \left(x\right) and h \left(x\right) on the same number plane.
Describe the transformation undergone by f \left(x\right) to become h \left(x\right).
For each of the following functions determine the function value when x = 2:
The x-intercepts of the graph of y = f \left( x \right) are x=- 5 and x=6. Find the x-intercepts of the following functions:
y = f \left( x + 4 \right)
y = f \left( x - 4 \right)
y = 3 f \left( x \right)
y = f \left( - x \right)
Consider the function f(x) = x^2-5. Find the following transformations in terms of x:
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
The graph of f \left(x\right) = 9^{x} and another exponential function, g \left(x\right) is shown:
g(x) increasing at exactly the same rate as f \left(x\right), but has a different y-intercept. Write down the equation of function g \left(x\right).