Use the accompanying graph of y=f(x) to sketch a graph of equation. (a) y=f(x+1)+1 (b) y=-f(x)-1

Use the accompanying graph of y=f(x) to sketch a graph of...

Key Concepts

Vertical Scaling (Stretching/Compression)

Vertical scaling modifies the height of the graph. Multiplying the function by a constant, as in y = a·f(x), scales the graph vertically. If |a| > 1, the graph stretches vertically, and if 0 < |a| < 1, it compresses vertically. This affects the steepness of the graph while maintaining its overall shape.

Horizontal Shifts

A horizontal shift moves a graph left or right without changing its shape. In the transformation y = f(x + c), where c is a constant, the graph shifts to the left by c units if c is positive and to the right by |c| units if c is negative.

Vertical Shifts

A vertical shift moves a graph up or down. For a transformation like y = f(x) + d, the graph is shifted upward by d units if d is positive and downward by |d| units if d is negative. This modification does not affect the shape of the graph.

Reflections

Reflections flip the graph over a specific axis. Reflecting over the x-axis, as in the transformation y = -f(x), inverts the values of f(x), producing a mirror image across the x-axis. Reflections can also occur across the y-axis when substituting x with -x.

Horizontal Scaling (Stretching/Compression)

Horizontal scaling alters the width of the graph. When the transformation involves replacing x with a constant multiple—such as in f(kx)—if k>1, the graph is horizontally compressed by a factor of 1/k, and if 0<k<1, it is stretched horizontally by a factor of 1/k.

Transcript

00:01 So in this problem, we're given the graph of y equals f of x, and we're asked to sketch a graph of each of the following equations.

00:08 So as you can see, i sketch the graph of y equals f of x as given in the book in red.

00:15 And so in part a, we're asked to sketch y equals f of x plus one plus one.

00:20 So let's start with f of x plus one.

00:24 So that just means we'll shift the graph of f of x to the left one unit.

00:30 So let's just shift each point to the left one.

00:41 And then after that we add one more, which just means that we add, we shift these points up by one unit.

00:52 So now let's just shift each of the blue points up by one.

00:56 So we get here, here, here, and finally here.

01:10 Okay, so now we can sketch through these points and that is what we get for part a.

01:26 Now part b asks us to sketch y equals negative f of x minus 1.

01:32 So let's start with negative f of x.

01:35 We'll reflect each point across the x axis.

01:40 So this one stays the same.

01:43 This one reflects down to here.

01:49 This one stays the same.

01:52 This one reflects up here, and then this one is here.

02:00 And just to make it clear, i'll cover up the ones that drew earlier in black.

02:14 Okay, so now we just want to shift each of these blue dots down one, since we're subtracting one.

02:22 So let's just shift each of them down one unit.

02:25 Unit to get here, here, here, here, and then here.

02:43 So then our sketch of part b, we just connect these points, and we get that our graph looks like this.

02:54 This will be getting part b...

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