Math  /  Algebra

QuestionGraph the given functions, f and g , in the same rectangular coordinate system. Then describe how the graph of g is related to the graph of f. f(x)=xg(x)=x6\begin{array}{l} f(x)=x \\ g(x)=x-6 \end{array}
Use the graphing tool to graph the functions.
How is the graph of f shifted to get the graph of g ? The graph of g is the graph of f shifted \square by \square units.

Studdy Solution

STEP 1

What is this asking? We're asked to graph two simple functions, f(x)=xf(x) = x and g(x)=x6g(x) = x - 6, and then describe how they relate to each other. Watch out! Don't mix up which way the graph shifts!
Subtracting a number shifts the graph *down*, not up!

STEP 2

1. Graph f(x)f(x)
2. Graph g(x)g(x)
3. Compare the graphs

STEP 3

Our first function is f(x)=xf(x) = x.
This is the simplest straight line we can think of!

STEP 4

Let's pick a few **key points** to plot this line.
If x=0x = 0, then f(0)=0f(0) = 0.
If x=1x = 1, then f(1)=1f(1) = 1.
And if x=1x = -1, then f(1)=1f(-1) = -1.

STEP 5

With these points, (0,0)(0, 0), (1,1)(1, 1), and (1,1)(-1, -1), we can draw a straight line through them.
It's a line that goes right through the **origin** and makes a perfect 45-degree angle with the x-axis!

STEP 6

Now, let's look at g(x)=x6g(x) = x - 6.
It looks almost the same as f(x)f(x), but with a little twist!

STEP 7

Again, let's pick some **smart points**.
If x=0x = 0, then g(0)=06=6g(0) = 0 - 6 = -6.
If x=6x = 6, then g(6)=66=0g(6) = 6 - 6 = 0.
And if x=1x = 1, then g(1)=16=5g(1) = 1 - 6 = -5.

STEP 8

Plot the points (0,6)(0, -6), (6,0)(6, 0), and (1,5)(1, -5).
Draw a straight line through them.
Notice anything?

STEP 9

Look closely at the two lines.
They're *parallel*, which means they have the same slope!
The only difference is their **vertical position**.

STEP 10

g(x)g(x) is just f(x)f(x) shifted *down* by **6 units**!
For any xx value, the g(x)g(x) value is **6 less** than the f(x)f(x) value.
That's exactly what subtracting 6 does!

STEP 11

The graph of gg is the graph of ff shifted *down* by **6** units.

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