Horizontal and vertical transformations are two of the many ways to convert the basic parent functions in a function family into their more complex counterparts.
What vertical and/or horizontal shifts must be applied to the parent function of \begin{align*}y = x^{2}\end{align*} in order to graph \begin{align*}g(x) = (x - 3)^{2} + 4\end{align*}?
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James Sousa - Function Transformations: Horizontal and Vertical Translations
Guidance
Have you ever tried to draw a picture of a rabbit, or cat, or dog? Unless you are talented, even the most common animals can be a bit of a challenge to draw accurately (or even recognizably!). One trick that can help even the most "artistically challenged" to create a clearly recognizable basic sketch is demonstrated in nearly all "learn to draw" courses: start with basic shapes. By starting your sketch with simple circles, ellipses, rectangles, etc., the basic outline of the more complex figure is easily arrived at, then details can be added as necessary, but the figure is already recognizable for what it is.
The same trick works when graphing equations. By learning the basic shapes of different types of function graphs, and then adjusting the graphs with different types of transformations, even complex graphs can be sketched rather easily. This lesson will focus on two particular types of transformations: vertical shifts and horizontal shifts.
We can express the application of vertical shifts this way:
- Formally: For any function f(x), the function g(x) = f(x) + c has a graph that is the same as f(x), shifted c units vertically. If c is positive, the graph is shifted up. If c is negative, the graph is shifted down.
- Informally: Adding a positive number after the x outside the parenthesis shifts the graph up, adding a negative (or subtracting) shifts the graph down.
We can express the application of horizontal shifts this way:
- Formally: given a function f(x), and a constant a > 0, the function g(x) = f(x - a) represents a horizontal shift a units to the right from f(x). The function h(x) = f(x + a) represents a horizontal shift a units to the left.
- Informally: Adding a positive number after the x inside the parenthesis shifts the graph left, adding a negative (or subtracting) shifts the graph right.
Example A
What must be done to the graph of y = x^{2} to convert it into the graphs of y = x^{2} - 3, and y = x^{2} + 4?
Solution:
At first glance, it may seem that the graphs have different widths. For example, it might look like y = x^{2} + 4, the uppermost of the three parabolas, is thinner than the other two parabolas. However, this is not the case. The parabolas are congruent.
If we shifted the graph of y = x^{2} up four units, we would have the exact same graph as y = x^{2} + 4. If we shifted y = x^{2} down three units, we would have the graph of y = x^{2} - 3.
Example B
Identify the transformation(s) involved in converting the graph of f(x) = |x| into g(x) = |x - 3|.
Solution:
From the examples of vertical shifts above, you might think that the graph of g(x) is the graph of f(x), shifted 3 units to the left. However, this is not the case. The graph of g(x) is the graph of f(x), shifted 3 units to the right.
The direction of the shift makes sense if we look at specific function values.
x | g(x) = abs(x - 3) |
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0 | 3 |
1 | 2 |
2 | 1 |
3 | 0 |
4 | 1 |
5 | 2 |
6 | 3 |
From the table we can see that the vertex of the graph is the point (3, 0). The function values on either side of x = 3 are symmetric, and greater than 0.
Example C
What transformations must be applied to \begin{align*}y = x^{2}\end{align*}, in order to graph \begin{align*}g(x) = (x + 2)^{2} - 2\end{align*}?
Solution
The graph of \begin{align*}g(x) = (x + 2)^{2} - 2\end{align*} is the graph of \begin{align*}y = x^{2}\end{align*} shifted 2 units to the left, and 2 units down.
Were you able to solve the question at the beginning of the lesson? "What transformations must be applied to \begin{align*}y = x^{2}\end{align*}, in order to graph \begin{align*}g(x) = (x - 3)^{2} + 4\end{align*}?" The graph of \begin{align*}g(x) = (x - 3)^{2} + 4\end{align*} is the graph of \begin{align*}y = x^{2}\end{align*} shifted 3 units to the right, and 4 units up. If you were able to identify the translation before the review, congratulations! You are on your way to an excellent conceptual base for manipulating functions. |
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Vocabulary
A shift, also known as a translation or a slide, is a transformation applied to the graph of a function which does not change the shape of the graph, only the location.
Vertical shifts are a result of adding a constant term to the value of a function. A positive term results in an upward shift, and a negative term in a downward shift.
Horizontal shifts are produced by adding a constant term to the function inside the parenthesis. A positive term results in a shift to the left and a negative term in a shift to the right (easily confused, pay attention!).
Guided Practice
Questions:
- 1) Use the graph of y = x^{2} to graph the function y = x^{2} - 5.
- 2) What is the relationship between f(x) = x^{2} and g(x) = (x - 2)^{2}?
- 3) What is the relationship between f(x) = x^{2} - 6 and f(x) = x^{2}?
- 4) Use the parent function f(x) = x^{2} to graph f(x) = x^{2} + 3.
- 5) Use the parent function f(x) = |x| to graph f(x) = |x - 4|.
Solutions:
- 1) The graph of y = x^{2} is a parabola with vertex at (0, 0).
- The graph of y = x^{2}- 5 is therefore a parabola with vertex (0, -5).
- To quickly sketch y = x^{2} - 5, you can sketch several points on y = x^{2}, and then shift them down 5 units.
- 2) The graph of g(x) is the graph of f(x), shifted 2 units to the right.
- 3) Adding or subtracting a value outside the parenthesis results in a vertical shift.
- Therefore, the graph of f(x) = x^{2} - 6 is the same as f(x) = x^{2} shifted 6 units down.
- 4) The function f(x) = x^{2} is a parabola with the vertex at (0, 0).
- As we saw in Q #3, adding outside the parenthesis shifts the graph vertically.
- Therefore, f(x) = x^{2} + 3 will be a parabola with the vertex 3 units up.
- 5) The graph of the absolute value function family parent function f(x) = |x| is a large "V" with the vertex at the origin.
- Adding or subtracting inside the parenthesis results in horizontal movement.
- Recall that the horizontal shift is right for negative numbers, and left for positive numbers.
- Therefore f(x) = |x - 4| is a large "V" with the vertex 4 units to the right of the origin.
Practice
- Graph the function \begin{align*}f(x) = 2 |x - 1| - 3\end{align*} without a calculator.
- What is the vertex of the graph and how do you know?
- Does it open up or down and how do you know?
- For the function: \begin{align*}f(x) = |x| + c\end{align*} if c is positive, the graph shifts in what direction?
- For the function: \begin{align*}f(x) = |x| + c\end{align*} if c is negative, the graph shifts in what direction?
- The function \begin{align*}g(x) = |x - a|\end{align*} represents a shift to the right or the left?
- The function \begin{align*}h(x) = |x + a|\end{align*} represents a shift to the right or the left?
- If a graph is in the form \begin{align*}a \cdot f(x)\end{align*}. What is the effect of changing the a?
Describe the transformation that has taken place for the parent function \begin{align*} f(x) = |x|\end{align*}
- \begin{align*}f(x) = |x| - 5\end{align*}
- \begin{align*}f(x) = 5|x + 7|\end{align*}
Write an equation that reflects the transformation that has taken place for the parent function \begin{align*} g(x) = \frac{1}{x}\end{align*}, for it to move in the following ways:
- Move two spaces up
- Move four spaces to the right
- Stretch it by 2 in the y-direction
Write an Equation for each described transformation.
- a V-shape shifted down 4 units.
- a V-shape shifted left 6 units
- a V-shape shifted right 2 units and up 1 unit.
The following graphs are transformations of the parent function \begin{align*}f(x) = |x|\end{align*} in the form of \begin{align*}f(x) = a|x - h| = k\end{align*}. Graph or sketch each to observe the type of transformation.
- \begin{align*}f(x) = |x| + 2\end{align*}. What happens to the graph when you add a number to the function? (i.e. f(x) + k).
- \begin{align*}f(x) = |x| - 4\end{align*}. What happens to the graph when you subtract a number from the function? (i.e. f(x) - k).
- \begin{align*}f(x) = |x - 4|\end{align*}. What happens to the graph when you subtract a number in the function? (i.e. f(x - h)).
- \begin{align*}f(x) = |x + 2|\end{align*}. What happens to the graph when you add a number in the function? (i.e. f(x + h)).
Practice: Graph the following:
- \begin{align*}f(x) = 2|x|\end{align*}
- \begin{align*}f(x) = \frac{5}{2}|x|\end{align*}
- \begin{align*}f(x) = \frac{1}{2}|x|\end{align*}
- \begin{align*}f(x) = \frac{2}{5}|x|\end{align*}
- Let \begin{align*}f(x) = x^2\end{align*}. Let \begin{align*}g(x)\end{align*} be the function obtained by shifting the graph of \begin{align*}f(x)\end{align*} two units to the right and then up three units. Find a formula for\begin{align*}g(x)\end{align*} and then draw its graph
Suppose H(t) gives the height of high tide in Hawaii(H) on a Tuesday, (t) of the year. Use shifts of the function H(t) to find formulas of each of the following functions:
- F(t), the height of high tide on Fiji on Tuesday (t), given that high tide in Fiji is always one foot higher than high tide in Hawaii.
- S(d), the height of high tide in Saint Thomas on Tuesday (t), given that high tide in Saint Thomas is the same height as the previous day's height in Hawaii.