Chapter Closure
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f(x) = 2x^2
Using this function we want to find the expression g(x) = f(x)+2. To do this, we will add 2 to the given function and simplify.Therefore, the transformed function g(x) is given by the following formula. g(x) = 2x^2+2 Vertical translations occur when we add or subtract a value from the output of a function. We can observe it more clearly in the table below.
| Vertical Translations | |
|---|---|
| Translation up k units, k>0 y=f(x)+ k | Translation down k units, k>0 y=f(x)- k |
Therefore, we can describe our transformation as vertical translation 2 units up. It means that the graph of the given function was shifted 2 units up.
f(x) = 2x^2
Using it, we want to find the expression h(x) = 2 * f(x). To do this, we will multiply the given function by 2 and simplify.Therefore, the transformed function h(x) is given by the following formula. h(x) = 4x^2 We can look at the following table to determine how we can describe out transformation.
| Vertical Stretch or Shrink | |
|---|---|
| Vertical stretch, a>1 y=af(x) | Vertical shrink, 0 |
We can describe our transformation as vertical stretch by a factor of 2.
f(x) = 2x^2
Using this function, we want to find the expression j(x) = f( x+2). To do this, we need to substitute x+2 for x in each instance of the x-variable.Therefore, the transformed function f(x) is given by the following formula. f(x) = 2(x+2)^2 Horizontal translations occur when we add or subtract a value from the input of a function. We can observe it more clearly in the table below.
| Horizontal Translations | |
|---|---|
| Translation right h units, h>0 y=f(x- h) | Translation left h units, h>0 y=f(x+ h) |
We can describe this transformation as horizontal translation 2 units left. It means that the graph of the given function was shifted 2 units left.
f(x) = 2x^2
Using the function f(x), we want to find the expression k(x)=f( 2x). To do this, we need to substitute 2x for x in each instance of the x-variable and simplify.x= 2x
(a * b)^m=a^m* b^m
Calculate power
Multiply
Therefore, the transformed function k(x) is given by the following formula. k(x) = 8x^2 Let's name our transformation using the table below.
| Horizontal Stretch or Shrink | |
|---|---|
| Horizontal stretch, 0 | Horizontal shrink, b>1 y=f(bx) |
We can describe this transformation as horizontal stretch by a factor of 2. However, we can look at the resulting function in different way — as a vertically stretched function f(x). k(x) = 8x^2 = 4(2x^2) ⇕ k(x) = 4 f(x) Hence, using the table below, we can also say that this transformation is a wertical stretch by factor of 4.
| Vertical Stretch or Shrink | |
|---|---|
| Vertical stretch, a>1 y=af(x) | Vertical shrink, 0 |