Answer The Following Questions.1. Function: $ T(p) = 7|p - 3| - 4 $- Vertex: - (3, -4)- Opens: - Up- Compared To The Parent Function: - NarrowerUse The Vertex And The Translations To Graph The Function.

Understanding the Function

The given function is in the form of $t(p) = 7|p - 3| - 4$. This is a linear function with a vertex at $(3, -4)$. The function opens upwards, indicating that it is a V-shaped graph. To graph this function, we need to understand the vertex and the translations involved.

Vertex Form of a Linear Function

The vertex form of a linear function is given by $f(x) = a| x - h| + k$, where $(h, k)$ is the vertex of the function. In this case, the vertex is $(3, -4)$, which means that the function has a minimum or maximum value at this point.

Translations in the Vertex Form

The vertex form of a linear function also involves translations. The translation in the x-direction is given by $h$, which is the x-coordinate of the vertex. The translation in the y-direction is given by $k$, which is the y-coordinate of the vertex.

Graphing the Function

To graph the function, we need to use the vertex and the translations. The vertex is at $(3, -4)$, which means that the function has a minimum value at this point. The function opens upwards, indicating that it is a V-shaped graph.

Step 1: Plot the Vertex

The first step in graphing the function is to plot the vertex at $(3, -4)$. This is the minimum point of the function.

Step 2: Determine the Slope

The slope of the function is determined by the coefficient of the absolute value term, which is $7$. This means that the function has a steep slope.

Step 3: Plot the Asymptotes

The asymptotes of the function are the lines that the function approaches as $x$ goes to infinity or negative infinity. In this case, the asymptotes are the lines $y = -4$ and $y = 4$.

Step 4: Plot the Graph

Using the vertex, slope, and asymptotes, we can plot the graph of the function. The graph will be a V-shaped graph that opens upwards.

Graphing the Function: A Step-by-Step Guide

Here is a step-by-step guide to graphing the function:

1. Plot the vertex at $(3, -4)$.
2. Determine the slope of the function, which is $7$.
3. Plot the asymptotes, which are the lines $y = -4$ and $y = 4$.
4. Plot the graph of the function using the vertex, slope, and asymptotes.

Example: Graphing the Function

Let's graph the function $t(p) = 7|p - 3| - 4$. We can use the steps outlined above to graph the function.

1. Plot the vertex at $(3, -4)$.
2. Determine the slope of the function, which is $7$.
3. Plot the asymptotes, which are the lines $y = -4$ and $y = 4$.
4. Plot the graph of the function using the vertex, slope, and asymptotes.

Conclusion

Graphing a linear function with vertex form involves understanding the vertex and the translations involved. The vertex form of a linear function is given by $f(x) = a| x - h| + k$, where $(h, k)$ is the vertex of the function. The translations in the vertex form are given by $h$ and $k$, which are the x and y coordinates of the vertex. By using the vertex, slope, and asymptotes, we can plot the graph of the function.

Key Takeaways

• The vertex form of a linear function is given by $f(x) = a| x - h| + k$, where $(h, k)$ is the vertex of the function.
• The translations in the vertex form are given by $h$ and $k$, which are the x and y coordinates of the vertex.
• To graph a linear function with vertex form, we need to plot the vertex, determine the slope, plot the asymptotes, and plot the graph using the vertex, slope, and asymptotes.

Frequently Asked Questions

• What is the vertex form of a linear function?
• How do I determine the slope of a linear function in vertex form?
• How do I plot the asymptotes of a linear function in vertex form?
• How do I plot the graph of a linear function in vertex form?

Answering the Questions

1. The vertex form of a linear function is given by $f(x) = a| x - h| + k$, where $(h, k)$ is the vertex of the function.
2. To determine the slope of a linear function in vertex form, we need to look at the coefficient of the absolute value term.
3. To plot the asymptotes of a linear function in vertex form, we need to plot the lines that the function approaches as $x$ goes to infinity or negative infinity.
4. To plot the graph of a linear function in vertex form, we need to use the vertex, slope, and asymptotes.
   Q&A: Graphing Linear Functions with Vertex Form =====================================================

Frequently Asked Questions

• What is the vertex form of a linear function?
• How do I determine the slope of a linear function in vertex form?
• How do I plot the asymptotes of a linear function in vertex form?
• How do I plot the graph of a linear function in vertex form?
• What is the significance of the vertex in a linear function?
• How do I find the vertex of a linear function?
• Can I use the vertex form to graph quadratic functions?
• How do I graph a linear function with a negative slope?
• Can I use the vertex form to graph absolute value functions?

Answering the Questions

Q1: What is the vertex form of a linear function?

A1: The vertex form of a linear function is given by $f(x) = a| x - h| + k$, where $(h, k)$ is the vertex of the function.

Q2: How do I determine the slope of a linear function in vertex form?

A2: To determine the slope of a linear function in vertex form, we need to look at the coefficient of the absolute value term. In the function $f(x) = a| x - h| + k$, the slope is given by $a$.

Q3: How do I plot the asymptotes of a linear function in vertex form?

A3: To plot the asymptotes of a linear function in vertex form, we need to plot the lines that the function approaches as $x$ goes to infinity or negative infinity. In the function $f(x) = a| x - h| + k$, the asymptotes are the lines $y = k - a$ and $y = k + a$.

Q4: How do I plot the graph of a linear function in vertex form?

A4: To plot the graph of a linear function in vertex form, we need to use the vertex, slope, and asymptotes. We can plot the vertex at $(h, k)$, draw a line with slope $a$ through the vertex, and plot the asymptotes.

Q5: What is the significance of the vertex in a linear function?

A5: The vertex of a linear function is the point where the function changes direction. It is the minimum or maximum point of the function.

Q6: How do I find the vertex of a linear function?

A6: To find the vertex of a linear function, we need to look at the vertex form of the function. In the function $f(x) = a| x - h| + k$, the vertex is given by $(h, k)$.

Q7: Can I use the vertex form to graph quadratic functions?

A7: No, the vertex form is used to graph linear functions, not quadratic functions. Quadratic functions have a different form, known as the standard form.

Q8: How do I graph a linear function with a negative slope?

A8: To graph a linear function with a negative slope, we need to use the vertex form of the function. We can plot the vertex at $(h, k)$, draw a line with slope $-a$ through the vertex, and plot the asymptotes.

Q9: Can I use the vertex form to graph absolute value functions?

A9: Yes, the vertex form can be used to graph absolute value functions. The vertex form of an absolute value function is given by $f(x) = a| x - h| + k$, where $(h, k)$ is the vertex of the function.

Conclusion

Graphing linear functions with vertex form involves understanding the vertex and the translations involved. The vertex form of a linear function is given by $f(x) = a| x - h| + k$, where $(h, k)$ is the vertex of the function. By using the vertex, slope, and asymptotes, we can plot the graph of the function.