Consider The Function $f(x)=-\frac{1}{4} X^2+x+3$.a. Evaluate F F F At X = − 4 , − 2 , 2 , 6 X = -4, -2, 2, 6 X = − 4 , − 2 , 2 , 6 , And 8 8 8 , And Present A Table With This Information.

$$

Introduction

In this article, we will be evaluating the quadratic function $f(x)=-\frac{1}{4} x^2+x+3$ at specific values of $x$. This will involve substituting the given values of $x$ into the function and calculating the corresponding values of $f(x)$. We will then present the results in a table for easy reference.

Evaluating the Function at Specific Values of $x$

To evaluate the function $f(x)=-\frac{1}{4} x^2+x+3$ at specific values of $x$, we will substitute the given values of $x$ into the function and simplify the resulting expression.

Evaluating $f(-4)$

To evaluate $f(-4)$, we will substitute $x=-4$ into the function:

$$f(-4)=-\frac{1}{4} (-4)^2+(-4)+3$$

Expanding and simplifying the expression, we get:

$$f(-4)=-\frac{1}{4} (16)+(-4)+3$$

$$f(-4)=-4+(-4)+3$$

$$f(-4)=-5$$

Evaluating $f(-2)$

To evaluate $f(-2)$, we will substitute $x=-2$ into the function:

$$f(-2)=-\frac{1}{4} (-2)^2+(-2)+3$$

Expanding and simplifying the expression, we get:

$$f(-2)=-\frac{1}{4} (4)+(-2)+3$$

$$f(-2)=-1+(-2)+3$$

$$f(-2)=0$$

Evaluating $f(2)$

To evaluate $f(2)$, we will substitute $x=2$ into the function:

$$f(2)=-\frac{1}{4} (2)^2+2+3$$

Expanding and simplifying the expression, we get:

$$f(2)=-\frac{1}{4} (4)+2+3$$

$$f(2)=-1+2+3$$

$$f(2)=4$$

Evaluating $f(6)$

To evaluate $f(6)$, we will substitute $x=6$ into the function:

$$f(6)=-\frac{1}{4} (6)^2+6+3$$

Expanding and simplifying the expression, we get:

$$f(6)=-\frac{1}{4} (36)+6+3$$

$$f(6)=-9+6+3$$

$$f(6)=0$$

Evaluating $f(8)$

To evaluate $f(8)$, we will substitute $x=8$ into the function:

$$f(8)=-\frac{1}{4} (8)^2+8+3$$

Expanding and simplifying the expression, we get:

$$f(8)=-\frac{1}{4} (64)+8+3$$

$$f(8)=-16+8+3$$

$$f(8)=-5$$

Presenting the Results in a Table

The following table presents the results of evaluating the function $f(x)=-\frac{1}{4} x^2+x+3$ at the specific values of $x$:

$x$	$f(x)$
-4	-5
-2	0
2	4
6	0
8	-5

Conclusion

In this article, we evaluated the quadratic function $f(x)=-\frac{1}{4} x^2+x+3$ at specific values of $x$. We presented the results in a table for easy reference. This type of evaluation is useful in a variety of mathematical and real-world applications, such as graphing functions and modeling real-world phenomena.

Future Directions

In future articles, we can explore other aspects of quadratic functions, such as graphing and modeling real-world phenomena. We can also investigate other types of functions, such as polynomial and rational functions.

References

• [1] "Quadratic Functions" by Math Open Reference
• [2] "Graphing Quadratic Functions" by Khan Academy

Glossary

• Quadratic function: A polynomial function of degree two, typically in the form $f(x)=ax^2+bx+c$.
• Graphing: The process of visualizing a function using a coordinate plane.
• Modeling: The process of using a mathematical function to describe a real-world phenomenon.
  

$$

Introduction

In our previous article, we evaluated the quadratic function $f(x)=-\frac{1}{4} x^2+x+3$ at specific values of $x$. In this article, we will answer some common questions related to quadratic functions and provide additional insights into the evaluation of $f(x)=-\frac{1}{4} x^2+x+3$.

Q&A

Q: What is a quadratic function?

A: A quadratic function is a polynomial function of degree two, typically in the form $f(x)=ax^2+bx+c$. Quadratic functions can be used to model a wide range of real-world phenomena, such as the trajectory of a projectile or the growth of a population.

Q: How do I evaluate a quadratic function at a specific value of $x$?

A: To evaluate a quadratic function at a specific value of $x$, you simply substitute the value of $x$ into the function and simplify the resulting expression. For example, to evaluate $f(x)=-\frac{1}{4} x^2+x+3$ at $x=2$, you would substitute $x=2$ into the function and get:

$$f(2)=-\frac{1}{4} (2)^2+2+3$$

Expanding and simplifying the expression, you get:

$$f(2)=-\frac{1}{4} (4)+2+3$$

$$f(2)=-1+2+3$$

$$f(2)=4$$

Q: What is the vertex of a quadratic function?

A: The vertex of a quadratic function is the point on the graph of the function where the function reaches its maximum or minimum value. The vertex can be found using the formula:

$$x=-\frac{b}{2a}$$

where $a$ and $b$ are the coefficients of the quadratic function.

Q: How do I find the vertex of $f(x)=-\frac{1}{4} x^2+x+3$?

A: To find the vertex of $f(x)=-\frac{1}{4} x^2+x+3$, we can use the formula:

$$x=-\frac{b}{2a}$$

In this case, $a=-\frac{1}{4}$ and $b=1$. Plugging these values into the formula, we get:

$$x=-\frac{1}{2(-\frac{1}{4})}$$

$$x=-\frac{1}{-\frac{1}{2}}$$

$$x=2$$

So, the vertex of $f(x)=-\frac{1}{4} x^2+x+3$ is at $x=2$.

Q: What is the axis of symmetry of a quadratic function?

A: The axis of symmetry of a quadratic function is a vertical line that passes through the vertex of the function. The equation of the axis of symmetry is:

$$x=-\frac{b}{2a}$$

Q: How do I find the axis of symmetry of $f(x)=-\frac{1}{4} x^2+x+3$?

A: To find the axis of symmetry of $f(x)=-\frac{1}{4} x^2+x+3$, we can use the formula:

$$x=-\frac{b}{2a}$$

In this case, $a=-\frac{1}{4}$ and $b=1$. Plugging these values into the formula, we get:

$$x=-\frac{1}{2(-\frac{1}{4})}$$

$$x=-\frac{1}{-\frac{1}{2}}$$

$$x=2$$

So, the axis of symmetry of $f(x)=-\frac{1}{4} x^2+x+3$ is the vertical line $x=2$.

Conclusion

In this article, we answered some common questions related to quadratic functions and provided additional insights into the evaluation of $f(x)=-\frac{1}{4} x^2+x+3$. We hope that this article has been helpful in understanding quadratic functions and their applications.

Future Directions

In future articles, we can explore other aspects of quadratic functions, such as graphing and modeling real-world phenomena. We can also investigate other types of functions, such as polynomial and rational functions.

References

• [1] "Quadratic Functions" by Math Open Reference
• [2] "Graphing Quadratic Functions" by Khan Academy

Glossary

• Quadratic function: A polynomial function of degree two, typically in the form $f(x)=ax^2+bx+c$.
• Graphing: The process of visualizing a function using a coordinate plane.
• Modeling: The process of using a mathematical function to describe a real-world phenomenon.