To sketch the graph of the function

$y = -2 \cdot 3^x + 6,$

we need to analyze its properties, including the asymptote, $y$-intercept, and $x$-intercept. Let's go step-by-step.

1. Asymptote

The given function is a transformed exponential function. The general form for an exponential function is:

$y = a \cdot b^x + c,$

where $c$ determines the horizontal asymptote. In our case:

$y = -2 \cdot 3^x + 6.$

As $x \to \infty$, the term $-2 \cdot 3^x \to -\infty$, but the horizontal shift $+6$ affects the graph's asymptote. Therefore, the horizontal asymptote is:

$y = 6.$

2. $y$-Intercept

The $y$-intercept is the point where the graph intersects the $y$-axis. This occurs when $x = 0$. Substitute $x = 0$ into the function:

$y = -2 \cdot 3^0 + 6 = -2 \cdot 1 + 6 = -2 + 6 = 4.$

So, the $y$-intercept is at:

$(0, 4).$

3. $x$-Intercept

The $x$-intercept is the point where the graph intersects the $x$-axis. This occurs when $y = 0$. Set the function equal to zero and solve for $x$:

$-2 \cdot 3^x + 6 = 0.$

Rearranging:

$-2 \cdot 3^x = -6,$

$3^x = \frac{-6}{-2} = 3.$

To solve for $x$, we use the logarithm:

$3^x = 3 \implies x = 1.$

So, the $x$-intercept is at:

$(1, 0).$

4. Sketch the Graph

• Asymptote: The horizontal asymptote is the line $y = 6$.
• Intercepts: The $y$-intercept is at $(0, 4)$ and the $x$-intercept is at $(1, 0)$.
• Behavior of the Graph: Since the coefficient of $3^x$ is negative, the graph will decrease exponentially as $x$ increases.

Summary of Key Points

• Equation of Asymptote: $y = 6$
• $y$-Intercept: $(0, 4)$
• $x$-Intercept: $(1, 0)$

With these points, you can sketch the graph. It starts near the asymptote $y = 6$, crosses the $y$-axis at $(0, 4)$, then the $x$-axis at $(1, 0)$, and continues decreasing.

Would you like a more detailed explanation or have any questions?

Related Questions:

1. How do you find the asymptotes of other types of functions, such as rational or logarithmic functions?
2. What is the impact of different transformations (like shifts or reflections) on the graph of an exponential function?
3. How do you calculate intercepts for functions involving different bases, like $2^x$ or $5^x$?
4. How do you sketch graphs of exponential decay functions, such as $y = 2 \cdot (1/3)^x + 4$?
5. What is the relationship between the base of the exponential function and the rate of growth or decay?

Tip: When sketching the graph of a function, always find the intercepts and asymptotes first; they provide a useful framework for understanding the graph's overall behavior.