### Assignments:

Unfinished Assignment Study Questions for Lesson 11

### Lesson Objectives:

- Graph a function by using transformations. Memorize the graphs of these basic functions, which can be considered building blocks for many other functions.

The identity function, y = x. This is a diagonal line passing through the origin.

The squaring function, y = x^2. This is a parabola, opening up, and it has the points, (0, 0), (1, 1), and (-1, 1).

The square root function, y = sqrt(x). This function goes through the points (0, 0) and (1, 1).

The cubing function, y = x^3. This goes through the points (0, 0), (1, 1), and (-1, -1).

The absolute value function, y = abs(x). This function has the points (0, 0), (1, 1), and (-1, 1) and looks like a 'v'.

The cube root function, y = root(3)(x). This function has the points (0, 0), (1, 1), and (1, -1).

And the reciprocal function, y = 1/x. This function doesn't touch the x or y-axis, and has the points (1, 1) and (-1, -1).

We can create graphs of new functions by shifting, stretching, shrinking and reflecting these graphs. For any function, y = f(x) and any positive constant, b:

If we add b to our function, then we shift up b units.

If we subtract b from our function, then we shift down b units.

If we add b inside of the parentheses to our input, then we're going to shift left b units.

And if we subtract b from the input, then we shift to the right b units. Describe how the graph of the function can be obtained from one of the basic graphs and then graph the function.

g(x) = x-5

So think of the graph of f(x) = x. This is a straight line through the origin. So since g(x) is the same as f(x)-5, then we need to shift the graph of f(x) down 5 units. So (1, 1) becomes (1, 1-5) and (0, 0) becomes (0, 0-5) and (-1, -1) becomes (-1, -1-5). Or (1, -4), (0, -5), and (-1, -6).

So we get the following graph. Describe how the graph of the function can be obtained from one of the basic graphs and then graph the function.

g(x) = (x+3)^2-1

Think of the graph of f(x) = x^2. It's a parabola opening up. Now g(x) is equivalent to f(x+3)-1. This means that we need to shift left 3 units and down 1 unit. Or in other words, subtract 3 from the x-coordinates and subtract 1 from the y-coordinates.

So (-1, 1) becomes (-4, 0) and (0, 0) becomes (-3, -1) and (1, 1) becomes (-2, 0). So here's the graph of g(x). For any function, y = f(x), the graph of y = -f(x) is the reflection of the graph of y = f(x) across the x-axis.

And the graph of y = f(-x) is the reflection of the graph of y = f(x) across the y-axis. For any function, y = f(x), and any constant b:

If we multiply b on the outside of the function, then we either stretch vertically, if abs(b) > 1 or shrink vertically if abs(b) is between 0 and 1. And we also reflect across the x-axis if b < 0.

Now if b is multiplied by our input, then we stretch horizontally if abs(b) is between 0 and 1, and shrink horizontally abs(b) > 1 and we also reflect across the y-axis if b < 0. A graph of y = f(x) is as follows.

Graph the given equation, g(x) = -3f(x).

So we're multiplying a negative number on the outside of f(x). So since -3 < 0, we reflect across the x-axis. And since the abs(-3) > 1, we also stretch vertically.

The new coordinates for g(x) can be obtained by multiplying all of our y-values by -3. So (-2, -3) becomes (-2, 9); and (1, 2) becomes (1, -6); and (0, 0) becomes (0, 0); and (3, 2) becomes (3, -6); and (6, 0) becomes (6, 0).

So here's the graph of g(x). The graph of y = f(x) is as follows.

Graph the given equation, g(x) = f(-x/2).

This is the same as g(x) = f(-1/2*x). Since we're multiplying our input by a negative value, that means that we're going to reflect the graph of f(x) across the y-axis. And since the abs(-1/2) is between 0 and 1, we're going to stretch horizontally.

The new coordinates for g(x) can be obtained from the coordinates of f(x) by dividing each of our x-values by -1/2 or multiplying each of our x-values by -2. So (-2, -3) becomes (4, -3); (1, 2) becomes (-2, 2); (0, 0) stays the same; (3, 2) becomes (-6, 2); and (6, 0) becomes (-12, 0).

So this is the graph of g(x). A graph of y = f(x) is as follows.

Graph the given equation, g(x) = (-f(x-1))/2+4 So we can rewrite this as g(x) = -1/2*f(x-1)+4. Since we're multiplying the outside of our function by -1/2, that means that because of the negative value, we're reflecting the graph of f(x) about the x-axis. And then since 1/2 is between 0 and 1, that means that we're shrinking vertically.

So let's start by multiplying all of our y-values by -1/2. So (-2, -3) becomes (-2, 3/2); (1, 2) becomes (1, -1); (0, 0) stays the same; (3, 2) becomes (3, -1); and (6, 0) stays the same.

Now since we're subtracting 1 on the inside of our parentheses, that means that we're shifting the graph of f(x) to the right one unit, or adding one to each of our x-values. So (-2, 3/2) becomes (-1, 3/2); (1, -1) becomes (2, -1); (0, 0) becomes (1, 0); (3, -1) becomes (4, -1); (6, 0) becomes (7, 0).

Then adding 4 to everything means that we're shifting up 4 units, or adding 4 to each of our y-coordinates. So (-1, 3/2) becomes (-1, 11/2); (2, -1) becomes (2, 3); (1, 0) becomes (1, 4); (4, -1) becomes (4, 3); and (7, 0) becomes (7,4). So this is the graph of g(x).