Math / Algebra

Question2 Mark for Review
Which of the following statements is true about the exponential function $h$ given by $h(x)=-3 \cdot 4^{x}$ ?
A $h$ is always increasing, and the graph of $h$ is always concave up. (B) $h$ is always increasing, and the graph of $h$ is always concave down. (C) $h$ is always decreasing, and the graph of $h$ is always concave up. (D) $h$ is always decreasing, and the graph of $h$ is always concave down.

Studdy Solution

STEP 1

What is this asking? Which statement correctly describes whether the function $h(x) = -3 \cdot 4^x$ is increasing or decreasing, and whether its graph is concave up or down? Watch out! Don't mix up increasing/decreasing with concave up/concave down!
They're related, but different.
Also, that negative sign in front can be tricky!

STEP 2

1. Analyze the function's behavior
2. Determine concavity

STEP 3

Let's investigate what happens as $x$ gets bigger.
Pick a couple of test values for $x$ , like $x = 1$ and $x = 2$ .

STEP 4

When $x = 1$ , $h(1) = -3 \cdot 4^1 = -3 \cdot 4 = -12$ .
Okay, cool!

STEP 5

When $x = 2$ , $h(2) = -3 \cdot 4^2 = -3 \cdot 16 = -48$ .
Notice how the function's value became more negative, meaning it decreased.

STEP 6

Let's try one more, just to be sure.
When $x=3$ , $h(3) = -3 \cdot 4^3 = -3 \cdot 64 = -192$ .
Yep, still decreasing!
So, $h(x)$ is always decreasing.

STEP 7

Concavity describes how the rate of change of the function changes.
Since $h(x)$ is always decreasing, let's look at how much it decreases between our test values.

STEP 8

Between $x=1$ and $x=2$ , the function changed by $-48 - (-12) = -36$ .
It decreased by 36.

STEP 9

Between $x=2$ and $x=3$ , the function changed by $-192 - (-48) = -144$ .
Woah, it decreased by 144 this time!

STEP 10

The rate at which the function is decreasing is getting bigger (more negative).
This means the graph is bending upwards, even though the function is decreasing.
So, the graph is concave up.
Think of it like a frown that's getting steeper!

STEP 11

Since $h$ is always decreasing, and the graph of $h$ is always concave up, the correct answer is (C).

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Studdy Solution

STEP 1

STEP 2

1. Analyze the function's behavior
2. Determine concavity

STEP 3

Let's investigate what happens as $x$ gets bigger.
Pick a couple of test values for $x$ , like $x = 1$ and $x = 2$ .

STEP 4

When $x = 1$ , $h(1) = -3 \cdot 4^1 = -3 \cdot 4 = -12$ .
Okay, cool!

STEP 5

When $x = 2$ , $h(2) = -3 \cdot 4^2 = -3 \cdot 16 = -48$ .
Notice how the function's value became more negative, meaning it decreased.

STEP 6

Let's try one more, just to be sure.
When $x=3$ , $h(3) = -3 \cdot 4^3 = -3 \cdot 64 = -192$ .
Yep, still decreasing!
So, $h(x)$ is always decreasing.

STEP 7

Concavity describes how the rate of change of the function changes.
Since $h(x)$ is always decreasing, let's look at how much it decreases between our test values.

STEP 8

Between $x=1$ and $x=2$ , the function changed by $-48 - (-12) = -36$ .
It decreased by 36.

STEP 9

Between $x=2$ and $x=3$ , the function changed by $-192 - (-48) = -144$ .
Woah, it decreased by 144 this time!

STEP 10

The rate at which the function is decreasing is getting bigger (more negative).
This means the graph is bending upwards, even though the function is decreasing.
So, the graph is concave up.
Think of it like a frown that's getting steeper!

STEP 11

Since $h$ is always decreasing, and the graph of $h$ is always concave up, the correct answer is (C).

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Studdy Solution

STEP 1

STEP 2

1. Analyze the function's behavior2. Determine concavity

STEP 3

Let's **investigate** what happens as xxx gets bigger.Pick a couple of **test values** for xxx, like x=1x = 1x=1 and x=2x = 2x=2.

STEP 4

When x=1x = 1x=1, h(1)=−3⋅41=−3⋅4=−12h(1) = -3 \cdot 4^1 = -3 \cdot 4 = -12h(1)=−3⋅41=−3⋅4=−12.Okay, cool!

STEP 5

When x=2x = 2x=2, h(2)=−3⋅42=−3⋅16=−48h(2) = -3 \cdot 4^2 = -3 \cdot 16 = -48h(2)=−3⋅42=−3⋅16=−48.Notice how the function's value became *more negative*, meaning it **decreased**.

STEP 6

Let's try one more, just to be sure.When x=3x=3x=3, h(3)=−3⋅43=−3⋅64=−192h(3) = -3 \cdot 4^3 = -3 \cdot 64 = -192h(3)=−3⋅43=−3⋅64=−192.Yep, still **decreasing**!So, h(x)h(x)h(x) is always **decreasing**.

STEP 7

Concavity describes how the *rate of change* of the function changes.Since h(x)h(x)h(x) is always decreasing, let's look at *how much* it decreases between our test values.

STEP 8

Between x=1x=1x=1 and x=2x=2x=2, the function changed by −48−(−12)=−36-48 - (-12) = -36−48−(−12)=−36.It decreased by **36**.

STEP 9

Between x=2x=2x=2 and x=3x=3x=3, the function changed by −192−(−48)=−144-192 - (-48) = -144−192−(−48)=−144.Woah, it decreased by **144** this time!

STEP 10

The *rate* at which the function is decreasing is getting *bigger* (more negative).This means the graph is bending *upwards*, even though the function is decreasing.So, the graph is **concave up**.Think of it like a frown that's getting steeper!

STEP 11

Since hhh is always **decreasing**, and the graph of hhh is always **concave up**, the correct answer is (C).

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1. Analyze the function's behavior
2. Determine concavity

Let's investigate what happens as $x$ gets bigger.
Pick a couple of test values for $x$ , like $x = 1$ and $x = 2$ .

When $x = 1$ , $h(1) = -3 \cdot 4^1 = -3 \cdot 4 = -12$ .
Okay, cool!

When $x = 2$ , $h(2) = -3 \cdot 4^2 = -3 \cdot 16 = -48$ .
Notice how the function's value became more negative, meaning it decreased.

Let's try one more, just to be sure.
When $x=3$ , $h(3) = -3 \cdot 4^3 = -3 \cdot 64 = -192$ .
Yep, still decreasing!
So, $h(x)$ is always decreasing.

Concavity describes how the rate of change of the function changes.
Since $h(x)$ is always decreasing, let's look at how much it decreases between our test values.

Between $x=1$ and $x=2$ , the function changed by $-48 - (-12) = -36$ .
It decreased by 36.

Between $x=2$ and $x=3$ , the function changed by $-192 - (-48) = -144$ .
Woah, it decreased by 144 this time!

The rate at which the function is decreasing is getting bigger (more negative).
This means the graph is bending upwards, even though the function is decreasing.
So, the graph is concave up.
Think of it like a frown that's getting steeper!

Since $h$ is always decreasing, and the graph of $h$ is always concave up, the correct answer is (C).