To graph the function $f(x) = \frac{3}{2} \cdot \left( \frac{4}{3} \right)^x - 1$, we can proceed as follows:

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Key Information:

1. Given Point: $x = 1, f(x) = 1$
2. Y-intercept: $f(0) = \frac{1}{2}$
3. Horizontal Asymptote: $y = -1$

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Steps to Plot the Function:

1. Y-Intercept at $x = 0$:
   Substituting $x = 0$ into the function:
   $f(0) = \frac{3}{2} \cdot \left( \frac{4}{3} \right)^0 - 1 = \frac{3}{2} \cdot 1 - 1 = \frac{3}{2} - 1 = \frac{1}{2}$
   Thus, the y-intercept is $\left( 0, \frac{1}{2} \right)$.

2. Point at $x = 1$:
   It is given that $f(1) = 1$, so the point $(1, 1)$ is on the graph.

3. Horizontal Asymptote:
   The function $f(x) = \frac{3}{2} \cdot \left( \frac{4}{3} \right)^x - 1$ has a horizontal asymptote at $y = -1$.
   This happens because the exponential term approaches 0 as $x \to -\infty$, leaving the constant $-1$ as the limiting value.

4. General Behavior of the Function:

   • The base $\frac{4}{3} > 1$, so the exponential function grows as $x$ increases.
   • As $x \to -\infty$, the function approaches the horizontal asymptote $y = -1$.

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Plotting Points:

• $(0, \frac{1}{2})$ → y-intercept
• $(1, 1)$ → another given point
• Horizontal asymptote: $y = -1$.

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Sketch:

1. Start with the horizontal asymptote $y = -1$. Draw this as a dashed horizontal line.
2. Plot the points $(0, \frac{1}{2})$ and $(1, 1)$.
3. Since the function grows exponentially, the curve will increase as $x \to \infty$ and approach $y = -1$ as $x \to -\infty$.

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Final Graph Description:

The graph starts close to $y = -1$ on the far left and increases through the points $(0, \frac{1}{2})$ and $(1, 1)$. It continues to grow exponentially as $x \to \infty$.

Would you like me to provide a visual graph of this function? Let me know if you need further explanations or calculations!

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Related Questions:

1. What happens to the function if the base $\frac{4}{3}$ is replaced with $2$?
2. How does the horizontal asymptote change if we replace $-1$ with another constant?
3. What is the behavior of the function as $x \to -\infty$?
4. How do you determine the y-intercept of any exponential function?
5. Can you rewrite the function in terms of a transformation of $y = a^x$?

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Tip:

The y-intercept of an exponential function $f(x) = a \cdot b^x + c$ can always be found by substituting $x = 0$.