Question

1. without graphing, state the following for the graph of the exponential function \\(f(x) = (7)^x\\) k3

a) y-intercepts:
b) domain:
c) range:
d) intervals of decrease:
e) intervals of increase:
f) equation of horizontal asymptote:

application

1. complete the following table. a6

function	does the function represent exponential growth or decay? explain your thought process.	initial value	growth or decay rate
\\(f(x) = 17(3)^x\\)

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 2,
"skills_matched": [
"step_cot",
"explain_then_answer",
"tutor"
],
"needs_drawing": false,
"concepts_used": [
"Exponent Rules"
],
"new_concepts": [
"Exponential Functions",
"Exponential Growth and Decay"
],
"current_concepts": [
"Exponent Rules",
"Exponential Functions",
"Exponential Growth and Decay"
]
}
</pre_analysis>

<reasoning>

Analyze the exponential function properties

Using the Exponent Rules knowledge point, we analyze the function \(f(x) = (7)^x\).

• The y-intercept occurs when \(x = 0\):

\[f(0) = 7^0 = 1\]
So, the y-intercept is \((0, 1)\).

• The domain of any basic exponential function \(f(x) = b^x\) (where \(b > 0\)) is all real numbers:

\[(-\infty, \infty) \quad \text{or} \quad \{x \in \mathbb{R}\}\]

• Since \(7^x > 0\) for all real \(x\), the range is:

\[(0, \infty) \quad \text{or} \quad \{y \in \mathbb{R} \mid y > 0\}\]

• Since the base \(b = 7 > 1\), the function is strictly increasing.
• Interval of decrease: None
• Interval of increase: \((-\infty, \infty)\)
• As \(x \to -\infty\), \(7^x \to 0\). Thus, the horizontal asymptote is:

\[y = 0\]

Analyze the first table row

We analyze the function \(f(x) = 315(0.87)^x\), which is in the form \(f(x) = a(b)^x\).

• Growth or Decay: Since the base \(b = 0.87\) is between \(0\) and \(1\) (\(0 < b < 1\)), it represents exponential decay.
• Initial Value: The initial value \(a\) is the value when \(x = 0\):

\[f(0) = 315(0.87)^0 = 315\]

• Growth or Decay Rate: The decay rate \(r\) is found using \(b = 1 - r\):

\[0.87 = 1 - r \implies r = 0.13 \quad (13\%)\]

Analyze the second table row

We analyze the function \(f(x) = 17(3)^x\), which is in the form \(f(x) = a(b)^x\).

• Growth or Decay: Since the base \(b = 3\) is greater than \(1\) (\(b > 1\)), it represents exponential growth.
• Initial Value: The initial value \(a\) is the value when \(x = 0\):

\[f(0) = 17(3)^0 = 17\]

• Growth or Decay Rate: The growth rate \(r\) is found using \(b = 1 + r\):

\[3 = 1 + r \implies r = 2 \quad (200\%)\]
</reasoning>

<answer>

Question 4

a) y-intercept: \((0, 1)\) (or \(y = 1\))
b) Domain: \((-\infty, \infty)\) (or \(\{x \in \mathbb{R}\}\))
c) Range: \((0, \infty)\) (or \(\{y \in \mathbb{R} \mid y > 0\}\))
d) Intervals of decrease: None
e) Intervals of increase: \((-\infty, \infty)\)
f) Equation of horizontal asymptote: \(y = 0\)

Question 5

FUNCTION	Does the function represent exponential growth or decay? Explain your thought process.	Initial value	Growth or decay rate
\(f(x) = 17(3)^x\)	Growth, because the base \(3\) is greater than \(1\).	\(17\)	\(200\%\) growth rate (since \(3 - 1 = 2\))

</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multi-part",
"knowledge_point": [
"Mathematics",
"Algebra",
"Exponential Functions"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 2,
"skills_matched": [
"step_cot",
"explain_then_answer",
"tutor"
],
"needs_drawing": false,
"concepts_used": [
"Exponent Rules"
],
"new_concepts": [
"Exponential Functions",
"Exponential Growth and Decay"
],
"current_concepts": [
"Exponent Rules",
"Exponential Functions",
"Exponential Growth and Decay"
]
}
</pre_analysis>

<reasoning>

Analyze the exponential function properties

Using the Exponent Rules knowledge point, we analyze the function \(f(x) = (7)^x\).

• The y-intercept occurs when \(x = 0\):

\[f(0) = 7^0 = 1\]
So, the y-intercept is \((0, 1)\).

• The domain of any basic exponential function \(f(x) = b^x\) (where \(b > 0\)) is all real numbers:

\[(-\infty, \infty) \quad \text{or} \quad \{x \in \mathbb{R}\}\]

• Since \(7^x > 0\) for all real \(x\), the range is:

\[(0, \infty) \quad \text{or} \quad \{y \in \mathbb{R} \mid y > 0\}\]

• Since the base \(b = 7 > 1\), the function is strictly increasing.
• Interval of decrease: None
• Interval of increase: \((-\infty, \infty)\)
• As \(x \to -\infty\), \(7^x \to 0\). Thus, the horizontal asymptote is:

\[y = 0\]

Analyze the first table row

We analyze the function \(f(x) = 315(0.87)^x\), which is in the form \(f(x) = a(b)^x\).

• Growth or Decay: Since the base \(b = 0.87\) is between \(0\) and \(1\) (\(0 < b < 1\)), it represents exponential decay.
• Initial Value: The initial value \(a\) is the value when \(x = 0\):

\[f(0) = 315(0.87)^0 = 315\]

• Growth or Decay Rate: The decay rate \(r\) is found using \(b = 1 - r\):

\[0.87 = 1 - r \implies r = 0.13 \quad (13\%)\]

Analyze the second table row

We analyze the function \(f(x) = 17(3)^x\), which is in the form \(f(x) = a(b)^x\).

• Growth or Decay: Since the base \(b = 3\) is greater than \(1\) (\(b > 1\)), it represents exponential growth.
• Initial Value: The initial value \(a\) is the value when \(x = 0\):

\[f(0) = 17(3)^0 = 17\]

• Growth or Decay Rate: The growth rate \(r\) is found using \(b = 1 + r\):

\[3 = 1 + r \implies r = 2 \quad (200\%)\]
</reasoning>

<answer>

Question 4

a) y-intercept: \((0, 1)\) (or \(y = 1\))
b) Domain: \((-\infty, \infty)\) (or \(\{x \in \mathbb{R}\}\))
c) Range: \((0, \infty)\) (or \(\{y \in \mathbb{R} \mid y > 0\}\))
d) Intervals of decrease: None
e) Intervals of increase: \((-\infty, \infty)\)
f) Equation of horizontal asymptote: \(y = 0\)

Question 5

FUNCTION	Does the function represent exponential growth or decay? Explain your thought process.	Initial value	Growth or decay rate
\(f(x) = 17(3)^x\)	Growth, because the base \(3\) is greater than \(1\).	\(17\)	\(200\%\) growth rate (since \(3 - 1 = 2\))

</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multi-part",
"knowledge_point": [
"Mathematics",
"Algebra",
"Exponential Functions"
]
}
</post_analysis>