Graph The Function F ( X ) = ( X + 1 ) ( X − 5 F(x)=(x+1)(x-5 F ( X ) = ( X + 1 ) ( X − 5 ]. Use The Steps Below To Complete The Graph:1. Identify The X X X -intercepts: ( − 1 , 0 (-1,0 ( − 1 , 0 ] And ( 5 , 0 (5,0 ( 5 , 0 ].2. Find The Midpoint Between The Intercepts (calculate, Don't Plot).3. Find The

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Introduction

Graphing functions is an essential skill in mathematics, and it's crucial to understand the steps involved in graphing a function. In this article, we will focus on graphing the function $f(x)=(x+1)(x-5)$. We will break down the process into manageable steps, making it easier to understand and visualize the graph.

Step 1: Identify the $x$-Intercepts

The first step in graphing the function $f(x)=(x+1)(x-5)$ is to identify the $x$-intercepts. The $x$-intercepts are the points where the graph of the function crosses the $x$-axis. To find the $x$-intercepts, we need to set the function equal to zero and solve for $x$.

f(x) = (x+1)(x-5) = 0

We can factor the expression to get:

(x+1)(x-5) = 0

This tells us that either $(x+1) = 0$ or $(x-5) = 0$. Solving for $x$, we get:

x+1 = 0 \Rightarrow x = -1

and

x-5 = 0 \Rightarrow x = 5

Therefore, the $x$-intercepts are $(-1,0)$ and $(5,0)$.

Step 2: Find the Midpoint between the Intercepts

The next step is to find the midpoint between the $x$-intercepts. The midpoint is the point that lies exactly halfway between the two intercepts. To find the midpoint, we need to calculate the average of the $x$-coordinates of the intercepts.

x_{mid} = \frac{x_1 + x_2}{2} = \frac{-1 + 5}{2} = 2

Therefore, the midpoint between the intercepts is $(2,0)$.

Step 3: Find the Vertex of the Parabola

The vertex of the parabola is the point that lies at the bottom of the parabola. To find the vertex, we need to use the formula:

x_{vertex} = \frac{-b}{2a}

In this case, $a = 1$ and $b = -6$. Plugging in the values, we get:

x_{vertex} = \frac{-(-6)}{2(1)} = 3

Therefore, the vertex of the parabola is $(3,0)$.

Step 4: Determine the Direction of the Parabola

The direction of the parabola is determined by the sign of the coefficient of the squared term. In this case, the coefficient is positive, which means that the parabola opens upward.

Step 5: Plot the Graph

Now that we have all the necessary information, we can plot the graph of the function. We start by plotting the $x$-intercepts, which are $(-1,0)$ and $(5,0)$. We then plot the midpoint between the intercepts, which is $(2,0)$. Finally, we plot the vertex of the parabola, which is $(3,0)$.

Conclusion

Graphing the function $f(x)=(x+1)(x-5)$ involves identifying the $x$-intercepts, finding the midpoint between the intercepts, finding the vertex of the parabola, determining the direction of the parabola, and plotting the graph. By following these steps, we can visualize the graph of the function and gain a deeper understanding of its behavior.

Key Takeaways

• The $x$-intercepts of the function $f(x)=(x+1)(x-5)$ are $(-1,0)$ and $(5,0)$.
• The midpoint between the intercepts is $(2,0)$.
• The vertex of the parabola is $(3,0)$.
• The parabola opens upward.
• The graph of the function can be plotted by using the $x$-intercepts, midpoint, and vertex.

Final Thoughts

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Introduction

Graphing functions is an essential skill in mathematics, and it's crucial to understand the steps involved in graphing a function. In our previous article, we walked through the steps to graph the function $f(x)=(x+1)(x-5)$. In this article, we will answer some of the most frequently asked questions about graphing this function.

Q: What are the $x$-intercepts of the function $f(x)=(x+1)(x-5)$?

A: The $x$-intercepts of the function $f(x)=(x+1)(x-5)$ are $(-1,0)$ and $(5,0)$. These are the points where the graph of the function crosses the $x$-axis.

Q: How do I find the midpoint between the $x$-intercepts?

A: To find the midpoint between the $x$-intercepts, you need to calculate the average of the $x$-coordinates of the intercepts. The formula for finding the midpoint is:

x_{mid} = \frac{x_1 + x_2}{2}

In this case, $x_1 = -1$ and $x_2 = 5$, so the midpoint is:

x_{mid} = \frac{-1 + 5}{2} = 2

Therefore, the midpoint between the intercepts is $(2,0)$.

Q: What is the vertex of the parabola?

A: The vertex of the parabola is the point that lies at the bottom of the parabola. To find the vertex, you need to use the formula:

x_{vertex} = \frac{-b}{2a}

In this case, $a = 1$ and $b = -6$, so the vertex is:

x_{vertex} = \frac{-(-6)}{2(1)} = 3

Therefore, the vertex of the parabola is $(3,0)$.

Q: Does the parabola open upward or downward?

A: The parabola opens upward because the coefficient of the squared term is positive. This means that the parabola will open upward and will have a minimum point at the vertex.

Q: How do I plot the graph of the function?

A: To plot the graph of the function, you need to start by plotting the $x$-intercepts, which are $(-1,0)$ and $(5,0)$. You then plot the midpoint between the intercepts, which is $(2,0)$. Finally, you plot the vertex of the parabola, which is $(3,0)$.

Q: What are some common mistakes to avoid when graphing the function?

A: Some common mistakes to avoid when graphing the function include:

• Not identifying the $x$-intercepts correctly
• Not finding the midpoint between the intercepts correctly
• Not plotting the vertex of the parabola correctly
• Not determining the direction of the parabola correctly

Conclusion

Graphing the function $f(x)=(x+1)(x-5)$ involves identifying the $x$-intercepts, finding the midpoint between the intercepts, finding the vertex of the parabola, determining the direction of the parabola, and plotting the graph. By following these steps and avoiding common mistakes, you can visualize the graph of the function and gain a deeper understanding of its behavior.

Key Takeaways

• The $x$-intercepts of the function $f(x)=(x+1)(x-5)$ are $(-1,0)$ and $(5,0)$.
• The midpoint between the intercepts is $(2,0)$.
• The vertex of the parabola is $(3,0)$.
• The parabola opens upward.
• The graph of the function can be plotted by using the $x$-intercepts, midpoint, and vertex.

Final Thoughts

Graphing functions is an essential skill in mathematics, and it's crucial to understand the steps involved in graphing a function. By following the steps outlined in this article and avoiding common mistakes, you can visualize the graph of the function $f(x)=(x+1)(x-5)$ and gain a deeper understanding of its behavior.