Math  /  Algebra

QuestionConsider the function. f(x)=x2+5f(x)=|x-2|+5
Which two statements about the graph of the function are true? For x<2x<2, the graph is decreasing, but for x>2x>2 the graph is increasing. For x<2x<2, the graph is increasing, but for x>2x>2 the graph is decreasing. The function has a maximum at (2,5)(2,5). The function has a maximum at (5,2)(5,2). The function has a minimum at (2,5)(2,5).

Studdy Solution

STEP 1

What is this asking? This problem wants us to figure out how the graph of this absolute value function behaves, specifically whether it increases or decreases on either side of x=2x=2, and whether it has a minimum or maximum at a particular point. Watch out! Don't mix up minimums and maximums!
Absolute value functions have a sharp point, and that's where the interesting stuff happens.

STEP 2

1. Analyze the function for x<2x < 2
2. Analyze the function for x>2x > 2
3. Find the minimum

STEP 3

Let's think about what happens when xx is less than **2**.
If xx is less than **2**, then x2x - 2 is *negative*.
For example, if x=1x = 1, then x2=12=1x - 2 = 1 - 2 = -1.

STEP 4

Because x2x - 2 is *negative* when x<2x < 2, the absolute value x2|x - 2| becomes (x2)-(x - 2), which simplifies to x+2-x + 2.
So, our function becomes f(x)=x+2+5f(x) = -x + 2 + 5, which simplifies to f(x)=x+7f(x) = -x + 7.

STEP 5

Now we can see that when x<2x < 2, the function is f(x)=x+7f(x) = -x + 7.
This is a line with a *negative* slope of 1-1, which means the graph is **decreasing** as xx gets bigger.

STEP 6

What happens when xx is greater than **2**?
In this case, x2x - 2 is *positive*.
For example, if x=3x = 3, then x2=32=1x - 2 = 3 - 2 = 1.

STEP 7

Since x2x - 2 is *positive* when x>2x > 2, the absolute value x2|x - 2| is just x2x - 2.
So, our function becomes f(x)=x2+5f(x) = x - 2 + 5, which simplifies to f(x)=x+3f(x) = x + 3.

STEP 8

We see that when x>2x > 2, the function is f(x)=x+3f(x) = x + 3.
This is a line with a *positive* slope of 11, meaning the graph is **increasing** as xx gets bigger.

STEP 9

We know that the graph decreases when x<2x < 2 and increases when x>2x > 2.
This means the function must have a **minimum** at x=2x = 2.

STEP 10

To find the *y*-coordinate of the minimum, we plug x=2x = 2 into the original function: f(2)=22+5=0+5=0+5=5f(2) = |2 - 2| + 5 = |0| + 5 = 0 + 5 = 5.

STEP 11

So, the function has a **minimum** at the point (2,5)(2, 5).

STEP 12

The two true statements are: "For x<2x < 2, the graph is decreasing, but for x>2x > 2 the graph is increasing" and "The function has a minimum at (2,5)(2, 5)".

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