Function F F F Is Defined By F ( X ) = − A X + B F(x) = -a^x + B F ( X ) = − A X + B , Where A A A And B B B Are Constants. In The X Y Xy X Y -plane, The Graph Of Y = F ( X ) − 15 Y = F(x) - 15 Y = F ( X ) − 15 Has A Y Y Y -intercept At $\left(0,

Introduction

In mathematics, functions are used to describe the relationship between variables. A function $f$ is defined by $f(x) = -a^x + b$, where $a$ and $b$ are constants. In this article, we will analyze the graph of $y = f(x) - 15$ and determine the $y$-intercept.

The Graph of $y = f(x) - 15$

The graph of $y = f(x) - 15$ is obtained by shifting the graph of $y = f(x)$ down by 15 units. This means that for every point $(x, y)$ on the graph of $y = f(x)$, the corresponding point on the graph of $y = f(x) - 15$ is $(x, y - 15)$.

The $y$-Intercept

The $y$-intercept of a graph is the point where the graph intersects the $y$-axis. In other words, it is the point where $x = 0$. To find the $y$-intercept of the graph of $y = f(x) - 15$, we need to find the value of $y$ when $x = 0$.

Finding the $y$-Intercept

Let's substitute $x = 0$ into the equation $y = f(x) - 15$. We get:

$$y = f(0) - 15$$

Since $f(x) = -a^x + b$, we have:

$$f(0) = -a^0 + b$$

Recall that $a^0 = 1$ for any nonzero number $a$. Therefore, we have:

$$f(0) = -1 + b$$

Substituting this into the equation for $y$, we get:

$$y = (-1 + b) - 15$$

Simplifying, we get:

$$y = -16 + b$$

Conclusion

In conclusion, the graph of $y = f(x) - 15$ has a $y$-intercept at $\left(0, -16 + b\right)$. This means that the graph intersects the $y$-axis at the point $\left(0, -16 + b\right)$.

Properties of the Graph

The graph of $y = f(x) - 15$ has several properties that are worth noting:

• The graph is a downward-opening parabola.
• The vertex of the parabola is at the point $\left(0, -16 + b\right)$.
• The graph intersects the $x$-axis at the point $\left(x, 0\right)$, where $x$ is a solution to the equation $-a^x + b = 15$.

Solving the Equation

To find the solution to the equation $-a^x + b = 15$, we can rearrange the equation to get:

$$-a^x = 15 - b$$

Dividing both sides by $-1$, we get:

$$a^x = b - 15$$

Taking the logarithm of both sides, we get:

$$x \log a = \log (b - 15)$$

Dividing both sides by $\log a$, we get:

$$x = \frac{\log (b - 15)}{\log a}$$

This is the solution to the equation $-a^x + b = 15$.

Graphing the Function

To graph the function $y = f(x) - 15$, we can use a graphing calculator or a computer algebra system. We can also use a table of values to plot the graph.

Table of Values

Here is a table of values for the function $y = f(x) - 15$:

$x$	$y$
0	-16 + b
1	-a + b - 15
2	a^2 - b - 15
3	-a^3 + b - 15
...	...

Graphing the Function

Using a graphing calculator or a computer algebra system, we can plot the graph of $y = f(x) - 15$. The graph will be a downward-opening parabola with a vertex at the point $\left(0, -16 + b\right)$.

Conclusion

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Q: What is the $y$-intercept of the graph of $y = f(x) - 15$?

A: The $y$-intercept of the graph of $y = f(x) - 15$ is the point where the graph intersects the $y$-axis. In other words, it is the point where $x = 0$. To find the $y$-intercept, we need to find the value of $y$ when $x = 0$. We get:

$$y = f(0) - 15$$

Since $f(x) = -a^x + b$, we have:

$$f(0) = -a^0 + b$$

Recall that $a^0 = 1$ for any nonzero number $a$. Therefore, we have:

$$f(0) = -1 + b$$

Substituting this into the equation for $y$, we get:

$$y = (-1 + b) - 15$$

Simplifying, we get:

$$y = -16 + b$$

So, the $y$-intercept of the graph of $y = f(x) - 15$ is $\left(0, -16 + b\right)$.

Q: What is the vertex of the graph of $y = f(x) - 15$?

A: The vertex of the graph of $y = f(x) - 15$ is the point where the graph changes from concave up to concave down. To find the vertex, we need to find the value of $x$ that makes the derivative of the function equal to zero. The derivative of the function $f(x) = -a^x + b$ is:

$$f'(x) = -a^x \log a$$

Setting the derivative equal to zero, we get:

$$-a^x \log a = 0$$

Since $a^x$ is never equal to zero, we can divide both sides by $-a^x$ to get:

$$\log a = 0$$

This is a contradiction, since $\log a$ is never equal to zero. Therefore, the graph of $y = f(x) - 15$ has no vertex.

Q: How do I graph the function $y = f(x) - 15$?

A: To graph the function $y = f(x) - 15$, you can use a graphing calculator or a computer algebra system. You can also use a table of values to plot the graph.

Here is a table of values for the function $y = f(x) - 15$:

$x$	$y$
0	-16 + b
1	-a + b - 15
2	a^2 - b - 15
3	-a^3 + b - 15
...	...

Using a graphing calculator or a computer algebra system, you can plot the graph of $y = f(x) - 15$. The graph will be a downward-opening parabola with a vertex at the point $\left(0, -16 + b\right)$.

Q: What is the solution to the equation $-a^x + b = 15$?

A: To find the solution to the equation $-a^x + b = 15$, we can rearrange the equation to get:

$$-a^x = 15 - b$$

Dividing both sides by $-1$, we get:

$$a^x = b - 15$$

Taking the logarithm of both sides, we get:

$$x \log a = \log (b - 15)$$

Dividing both sides by $\log a$, we get:

$$x = \frac{\log (b - 15)}{\log a}$$

This is the solution to the equation $-a^x + b = 15$.

Q: What is the relationship between the graph of $y = f(x) - 15$ and the graph of $y = f(x)$?

A: The graph of $y = f(x) - 15$ is obtained by shifting the graph of $y = f(x)$ down by 15 units. This means that for every point $(x, y)$ on the graph of $y = f(x)$, the corresponding point on the graph of $y = f(x) - 15$ is $(x, y - 15)$.

Q: What is the significance of the $y$-intercept of the graph of $y = f(x) - 15$?

A: The $y$-intercept of the graph of $y = f(x) - 15$ is the point where the graph intersects the $y$-axis. This point is significant because it represents the value of the function when $x = 0$. In other words, it represents the value of the function at the origin.

Q: How do I determine the values of $a$ and $b$ in the function $f(x) = -a^x + b$?

A: To determine the values of $a$ and $b$ in the function $f(x) = -a^x + b$, you need to have additional information about the function. For example, you may know the value of the function at a particular point, or you may know the derivative of the function at a particular point. Using this information, you can solve for the values of $a$ and $b$.

Conclusion

In conclusion, the graph of $y = f(x) - 15$ has a $y$-intercept at $\left(0, -16 + b\right)$. The graph is a downward-opening parabola with a vertex at the point $\left(0, -16 + b\right)$. The graph intersects the $x$-axis at the point $\left(x, 0\right)$, where $x$ is a solution to the equation $-a^x + b = 15$.