Express The Function \[$ F(x) \$\] As Follows:$\[ F(x) = \frac{3}{4}x - 2 \\]

Introduction

In mathematics, a linear function is a polynomial function of degree one, which can be written in the form $f(x) = mx + b$, where $m$ is the slope and $b$ is the y-intercept. In this article, we will explore how to express the linear function $f(x) = \frac{3}{4}x - 2$ in different forms.

Standard Form

The standard form of a linear function is $f(x) = mx + b$, where $m$ is the slope and $b$ is the y-intercept. In this case, the slope is $\frac{3}{4}$ and the y-intercept is $-2$. Therefore, the standard form of the function is:

$f(x) = \frac{3}{4}x - 2$

Slope-Intercept Form

The slope-intercept form of a linear function is $f(x) = mx + b$, where $m$ is the slope and $b$ is the y-intercept. In this case, the slope is $\frac{3}{4}$ and the y-intercept is $-2$. Therefore, the slope-intercept form of the function is:

$f(x) = \frac{3}{4}x - 2$

Point-Slope Form

The point-slope form of a linear function is $f(x) - f(a) = m(x - a)$, where $m$ is the slope and $(a, f(a))$ is a point on the line. In this case, we can choose any point on the line, such as $(0, -2)$. Therefore, the point-slope form of the function is:

$f(x) - (-2) = \frac{3}{4}(x - 0)$

Simplifying the equation, we get:

$f(x) = \frac{3}{4}x - 2$

General Form

The general form of a linear function is $ax + by = c$, where $a$, $b$, and $c$ are constants. In this case, we can rewrite the function as:

$\frac{3}{4}x - 2 = 0$

Multiplying both sides by $4$, we get:

$3x - 8 = 0$

Therefore, the general form of the function is:

$3x - 8 = 0$

Vertex Form

The vertex form of a linear function is $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. In this case, the vertex is $(0, -2)$. Therefore, the vertex form of the function is:

$f(x) = \frac{3}{4}(x - 0)^2 - 2$

Simplifying the equation, we get:

$f(x) = \frac{3}{4}x^2 - 2$

Conclusion

In this article, we have explored how to express the linear function $f(x) = \frac{3}{4}x - 2$ in different forms. We have seen that the standard form, slope-intercept form, point-slope form, general form, and vertex form are all equivalent and can be used to represent the same function. By understanding these different forms, we can better analyze and solve linear equations.

Key Takeaways

• The standard form of a linear function is $f(x) = mx + b$.
• The slope-intercept form of a linear function is $f(x) = mx + b$.
• The point-slope form of a linear function is $f(x) - f(a) = m(x - a)$.
• The general form of a linear function is $ax + by = c$.
• The vertex form of a linear function is $f(x) = a(x - h)^2 + k$.

References

• [1] "Linear Functions" by Math Open Reference. Retrieved February 2023.
• [2] "Slope-Intercept Form" by Khan Academy. Retrieved February 2023.
• [3] "Point-Slope Form" by Purplemath. Retrieved February 2023.
• [4] "General Form" by Mathway. Retrieved February 2023.
• [5] "Vertex Form" by IXL. Retrieved February 2023.
  Q&A: Expressing a Linear Function in Different Forms =====================================================

Introduction

In our previous article, we explored how to express the linear function $f(x) = \frac{3}{4}x - 2$ in different forms. In this article, we will answer some frequently asked questions about expressing linear functions in different forms.

Q: What is the standard form of a linear function?

A: The standard form of a linear function is $f(x) = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Q: What is the slope-intercept form of a linear function?

A: The slope-intercept form of a linear function is $f(x) = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Q: What is the point-slope form of a linear function?

A: The point-slope form of a linear function is $f(x) - f(a) = m(x - a)$, where $m$ is the slope and $(a, f(a))$ is a point on the line.

Q: What is the general form of a linear function?

A: The general form of a linear function is $ax + by = c$, where $a$, $b$, and $c$ are constants.

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

A: The vertex form of a linear function is $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.

Q: How do I convert a linear function from one form to another?

A: To convert a linear function from one form to another, you can use the following steps:

• Standard form to slope-intercept form: $f(x) = mx + b$
• Slope-intercept form to standard form: $f(x) = mx + b$
• Point-slope form to standard form: $f(x) - f(a) = m(x - a)$
• General form to standard form: $ax + by = c$
• Vertex form to standard form: $f(x) = a(x - h)^2 + k$

Q: What are some common mistakes to avoid when expressing a linear function in different forms?

A: Some common mistakes to avoid when expressing a linear function in different forms include:

• Not using the correct slope and y-intercept in the standard form
• Not using the correct point on the line in the point-slope form
• Not using the correct constants in the general form
• Not using the correct vertex in the vertex form

Q: How do I determine which form to use when expressing a linear function?

A: The form you use to express a linear function will depend on the specific problem you are trying to solve. For example, if you are given a point on the line and the slope, you may want to use the point-slope form. If you are given the y-intercept and the slope, you may want to use the slope-intercept form.

Conclusion

In this article, we have answered some frequently asked questions about expressing linear functions in different forms. We hope that this article has been helpful in clarifying some of the common mistakes and misconceptions about expressing linear functions in different forms.

Key Takeaways

• The standard form of a linear function is $f(x) = mx + b$.
• The slope-intercept form of a linear function is $f(x) = mx + b$.
• The point-slope form of a linear function is $f(x) - f(a) = m(x - a)$.
• The general form of a linear function is $ax + by = c$.
• The vertex form of a linear function is $f(x) = a(x - h)^2 + k$.

References

• [1] "Linear Functions" by Math Open Reference. Retrieved February 2023.
• [2] "Slope-Intercept Form" by Khan Academy. Retrieved February 2023.
• [3] "Point-Slope Form" by Purplemath. Retrieved February 2023.
• [4] "General Form" by Mathway. Retrieved February 2023.
• [5] "Vertex Form" by IXL. Retrieved February 2023.