The Table Of Values Shows A Quadratic Function.${ \begin{tabular}{|c|c|c|c|c|c|} \hline X X X & -4 & -3 & -1 & 1 & 2 \ \hline F ( X ) F(x) F ( X ) & 4 & -6 & -14 & -6 & 4 \ \hline \end{tabular} }$What Is The Equation Of The Function?A. $f(x) =

Understanding Quadratic Functions

A quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. It can be written in the form of $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. The graph of a quadratic function is a parabola, which is a U-shaped curve. In this article, we will discuss how to find the equation of a quadratic function given its table of values.

Analyzing the Table of Values

The table of values shows the input values of $x$ and the corresponding output values of $f(x)$. We can see that the function has a symmetry about the line $x = 1$. This means that the function is an even function, and we can write it in the form of $f(x) = a(x - 1)^2 + k$, where $a$ and $k$ are constants.

Finding the Equation of the Function

To find the equation of the function, we need to find the values of $a$ and $k$. We can do this by using the given table of values. Let's start by finding the value of $a$. We can use the fact that the function is even, and the symmetry about the line $x = 1$. This means that the value of $f(x)$ at $x = -1$ is equal to the value of $f(x)$ at $x = 1$. We can write this as:

$$f(-1) = f(1)$$

Substituting the values from the table of values, we get:

$$-14 = -6$$

This is not true, so we need to find another way to find the value of $a$. Let's use the fact that the function is quadratic, and the graph of a quadratic function is a parabola. We can write the equation of the function in the form of $f(x) = ax^2 + bx + c$. We can use the given table of values to find the values of $a$, $b$, and $c$.

Using the Table of Values to Find the Equation

We can use the table of values to find the values of $a$, $b$, and $c$. Let's start by finding the value of $a$. We can use the fact that the function is quadratic, and the graph of a quadratic function is a parabola. We can write the equation of the function in the form of $f(x) = ax^2 + bx + c$. We can use the given table of values to find the values of $a$, $b$, and $c$.

Let's use the first row of the table of values to find the value of $a$. We can write the equation of the function as:

$$f(x) = ax^2 + bx + c$$

Substituting the values from the first row of the table of values, we get:

$$4 = a(-4)^2 + b(-4) + c$$

Simplifying the equation, we get:

$$4 = 16a - 4b + c$$

Now, let's use the second row of the table of values to find the value of $b$. We can write the equation of the function as:

$$f(x) = ax^2 + bx + c$$

Substituting the values from the second row of the table of values, we get:

$$-6 = a(-3)^2 + b(-3) + c$$

Simplifying the equation, we get:

$$-6 = 9a - 3b + c$$

Now, let's use the third row of the table of values to find the value of $c$. We can write the equation of the function as:

$$f(x) = ax^2 + bx + c$$

Substituting the values from the third row of the table of values, we get:

$$-14 = a(-1)^2 + b(-1) + c$$

Simplifying the equation, we get:

$$-14 = a - b + c$$

Now, we have three equations and three unknowns. We can solve the system of equations to find the values of $a$, $b$, and $c$.

Solving the System of Equations

We can solve the system of equations using substitution or elimination. Let's use substitution. We can solve the first equation for $c$:

$$c = 4 - 16a + 4b$$

Substituting this expression for $c$ into the second equation, we get:

$$-6 = 9a - 3b + 4 - 16a + 4b$$

Simplifying the equation, we get:

$$-10 = -7a + b$$

Now, let's solve the third equation for $c$:

$$c = -14 - a + b$$

Substituting this expression for $c$ into the first equation, we get:

$$4 = 16a - 4b - 14 - a + b$$

Simplifying the equation, we get:

$$18 = 15a - 3b$$

Now, we have two equations and two unknowns. We can solve the system of equations to find the values of $a$ and $b$.

Finding the Values of $a$ and $b$

We can solve the system of equations using substitution or elimination. Let's use substitution. We can solve the first equation for $b$:

$$b = -10 + 7a$$

Substituting this expression for $b$ into the second equation, we get:

$$18 = 15a - 3(-10 + 7a)$$

Simplifying the equation, we get:

$$18 = 15a + 30 - 21a$$

Simplifying the equation further, we get:

$$-12 = -6a$$

Dividing both sides of the equation by $-6$, we get:

$$a = 2$$

Now, let's find the value of $b$. We can substitute the value of $a$ into the expression for $b$:

$$b = -10 + 7(2)$$

Simplifying the equation, we get:

$$b = -10 + 14$$

Simplifying the equation further, we get:

$$b = 4$$

Now, we have the values of $a$ and $b$. We can substitute these values into the expression for $c$:

$$c = 4 - 16(2) + 4(4)$$

Simplifying the equation, we get:

$$c = 4 - 32 + 16$$

Simplifying the equation further, we get:

$$c = -12$$

Now, we have the values of $a$, $b$, and $c$. We can write the equation of the function as:

$$f(x) = 2x^2 + 4x - 12$$

Conclusion

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Frequently Asked Questions

In this article, we will answer some frequently asked questions about quadratic functions.

Q: What is a quadratic function?

A: A quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. It can be written in the form of $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

Q: What is the graph of a quadratic function?

A: The graph of a quadratic function is a parabola, which is a U-shaped curve.

Q: How do I find the equation of a quadratic function given its table of values?

A: To find the equation of a quadratic function given its table of values, you can use the fact that the function is even, and the symmetry about the line $x = 1$. You can also use the fact that the function is quadratic, and the graph of a quadratic function is a parabola. You can write the equation of the function in the form of $f(x) = ax^2 + bx + c$, and use the given table of values to find the values of $a$, $b$, and $c$.

Q: How do I solve a system of equations to find the values of $a$, $b$, and $c$?

A: To solve a system of equations, you can use substitution or elimination. Let's use substitution. You can solve one equation for one variable, and then substitute that expression into the other equation. You can repeat this process until you have found the values of all the variables.

Q: What is the vertex of a quadratic function?

A: The vertex of a quadratic function is the point on the graph where the function changes from decreasing to increasing, or from increasing to decreasing. It is the lowest or highest point on the graph.

Q: How do I find the vertex of a quadratic function?

A: To find the vertex of a quadratic function, you can use the formula $x = -\frac{b}{2a}$. This will give you the x-coordinate of the vertex. You can then substitute this value into the equation of the function to find the y-coordinate of the vertex.

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

A: The axis of symmetry of a quadratic function is the vertical line that passes through the vertex of the function. It is the line of symmetry about which the graph of the function is reflected.

Q: How do I find the axis of symmetry of a quadratic function?

A: To find the axis of symmetry of a quadratic function, you can use the formula $x = -\frac{b}{2a}$. This will give you the x-coordinate of the axis of symmetry.

Q: What is the domain and range of a quadratic function?

A: The domain of a quadratic function is the set of all possible input values of the function. The range of a quadratic function is the set of all possible output values of the function.

Q: How do I find the domain and range of a quadratic function?

A: To find the domain and range of a quadratic function, you can look at the graph of the function. The domain is the set of all x-values that are on the graph, and the range is the set of all y-values that are on the graph.

Conclusion

In this article, we have answered some frequently asked questions about quadratic functions. We have discussed the definition of a quadratic function, the graph of a quadratic function, and how to find the equation of a quadratic function given its table of values. We have also discussed how to solve a system of equations to find the values of $a$, $b$, and $c$, and how to find the vertex, axis of symmetry, domain, and range of a quadratic function.