The Table Describes The Quadratic Function $p(x)$. \[ \begin{tabular}{|l|l|} \hline X$ & P ( X ) P(x) P ( X ) \ \hline -1 & 31 \ \hline 0 & 17 \ \hline 1 & 7 \ \hline 2 & 1 \ \hline 3 & -1 \ \hline 4 & 1 \ \hline 5 & 7

The Table Describes the Quadratic Function p(x)

Understanding Quadratic Functions

Quadratic functions are a type of polynomial function that can be written in the form of $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. These functions are characterized by their parabolic shape, which can either open upwards or downwards. In this article, we will explore the quadratic function $p(x)$, which is described by the table below.

The Table Describes the Quadratic Function p(x)

$x$	$p(x)$
-1	31
0	17
1	7
2	1
3	-1
4	1
5	7

Analyzing the Table

From the table, we can see that the function $p(x)$ takes on different values for different inputs of $x$. We can start by analyzing the pattern of the function. As we can see, the function increases from $x = -1$ to $x = 0$, then decreases from $x = 0$ to $x = 1$, and continues to decrease until $x = 3$. After that, the function increases again until $x = 5$.

Finding the Quadratic Function

To find the quadratic function $p(x)$, we need to determine the values of $a$, $b$, and $c$. We can do this by using the data points from the table and substituting them into the general form of a quadratic function. Let's start by substituting the data point $(0, 17)$ into the function:

$$17 = a(0)^2 + b(0) + c$$

This simplifies to:

$$17 = c$$

So, we know that $c = 17$. Now, let's substitute the data point $(1, 7)$ into the function:

$$7 = a(1)^2 + b(1) + 17$$

This simplifies to:

$$-10 = a + b$$

Now, let's substitute the data point $(2, 1)$ into the function:

$$1 = a(2)^2 + b(2) + 17$$

This simplifies to:

$$-27 = 4a + 2b$$

We now have a system of two equations with two unknowns:

$$-10 = a + b$$

$$-27 = 4a + 2b$$

We can solve this system of equations by multiplying the first equation by 2 and subtracting it from the second equation:

$$-20 = 2a + 2b$$

$$-27 = 4a + 2b$$

Subtracting the first equation from the second equation, we get:

$$-7 = 2a$$

Dividing both sides by 2, we get:

$$a = -\frac{7}{2}$$

Now that we know the value of $a$, we can substitute it into one of the original equations to find the value of $b$. Let's use the first equation:

$$-10 = a + b$$

Substituting $a = -\frac{7}{2}$, we get:

$$-10 = -\frac{7}{2} + b$$

Adding $\frac{7}{2}$ to both sides, we get:

$$b = -\frac{23}{2}$$

Now that we know the values of $a$ and $b$, we can write the quadratic function $p(x)$:

$$p(x) = -\frac{7}{2}x^2 - \frac{23}{2}x + 17$$

Graphing the Quadratic Function

To graph the quadratic function $p(x)$, we can use the data points from the table. We can plot the points $(x, p(x))$ for each value of $x$ in the table. We can then connect the points to form a parabola.

Conclusion

In this article, we analyzed the quadratic function $p(x)$, which is described by the table below. We found the values of $a$, $b$, and $c$ by using the data points from the table and substituting them into the general form of a quadratic function. We then wrote the quadratic function $p(x)$ and graphed it using the data points from the table. The graph of the function is a parabola that opens downwards.

Future Work

In the future, we can explore other types of quadratic functions and analyze their properties. We can also use the quadratic function $p(x)$ to model real-world phenomena, such as the motion of an object under the influence of gravity.

References

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

Glossary

• Quadratic function: A type of polynomial function that can be written in the form of $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.
• Parabola: A type of curve that is formed by a quadratic function.
• Data point: A point on a graph that represents a value of the function at a particular input.

Keywords

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• Parabola
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  Quadratic Function p(x) Q&A

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about the quadratic function $p(x)$.

Q: What is the quadratic function p(x)?

A: The quadratic function $p(x)$ is a type of polynomial function that 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 the quadratic function p(x)?

A: The graph of the quadratic function $p(x)$ is a parabola that opens downwards.

Q: How do I find the values of a, b, and c for the quadratic function p(x)?

A: To find the values of $a$, $b$, and $c$ for the quadratic function $p(x)$, you can use the data points from the table and substitute them into the general form of a quadratic function.

Q: How do I graph the quadratic function p(x)?

A: To graph the quadratic function $p(x)$, you can use the data points from the table and plot the points $(x, p(x))$ for each value of $x$ in the table. You can then connect the points to form a parabola.

Q: What is the vertex of the quadratic function p(x)?

A: The vertex of the quadratic function $p(x)$ is the point on the graph where the parabola changes direction. To find the vertex, you can use the formula $x = -\frac{b}{2a}$.

Q: How do I find the x-intercepts of the quadratic function p(x)?

A: To find the x-intercepts of the quadratic function $p(x)$, you can set $p(x) = 0$ and solve for $x$.

Q: How do I find the y-intercept of the quadratic function p(x)?

A: To find the y-intercept of the quadratic function $p(x)$, you can set $x = 0$ and solve for $p(x)$.

Q: What is the domain of the quadratic function p(x)?

A: The domain of the quadratic function $p(x)$ is all real numbers.

Q: What is the range of the quadratic function p(x)?

A: The range of the quadratic function $p(x)$ is all real numbers.

Q: How do I determine if the quadratic function p(x) is increasing or decreasing?

A: To determine if the quadratic function $p(x)$ is increasing or decreasing, you can look at the coefficient of the $x^2$ term. If the coefficient is positive, the function is decreasing. If the coefficient is negative, the function is increasing.

Q: How do I find the axis of symmetry of the quadratic function p(x)?

A: To find the axis of symmetry of the quadratic function $p(x)$, you can use the formula $x = -\frac{b}{2a}$.

Q: How do I find the vertex form of the quadratic function p(x)?

A: To find the vertex form of the quadratic function $p(x)$, you can use the formula $p(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.

Q: How do I find the standard form of the quadratic function p(x)?

A: To find the standard form of the quadratic function $p(x)$, you can use the formula $p(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

Q: How do I find the quadratic function p(x) from a given table of values?

A: To find the quadratic function $p(x)$ from a given table of values, you can use the data points from the table and substitute them into the general form of a quadratic function.

Q: How do I find the quadratic function p(x) from a given graph?

A: To find the quadratic function $p(x)$ from a given graph, you can use the points on the graph and substitute them into the general form of a quadratic function.

Q: How do I find the quadratic function p(x) from a given equation?

A: To find the quadratic function $p(x)$ from a given equation, you can use the equation and substitute it into the general form of a quadratic function.

Q: How do I find the quadratic function p(x) from a given set of x and y values?

A: To find the quadratic function $p(x)$ from a given set of x and y values, you can use the data points from the table and substitute them into the general form of a quadratic function.

Q: How do I find the quadratic function p(x) from a given set of x and y coordinates?

A: To find the quadratic function $p(x)$ from a given set of x and y coordinates, you can use the data points from the table and substitute them into the general form of a quadratic function.

Q: How do I find the quadratic function p(x) from a given set of x and y values with errors?

A: To find the quadratic function $p(x)$ from a given set of x and y values with errors, you can use the data points from the table and substitute them into the general form of a quadratic function, taking into account the errors in the data.

Q: How do I find the quadratic function p(x) from a given set of x and y values with outliers?

A: To find the quadratic function $p(x)$ from a given set of x and y values with outliers, you can use the data points from the table and substitute them into the general form of a quadratic function, taking into account the outliers in the data.

Q: How do I find the quadratic function p(x) from a given set of x and y values with missing values?

A: To find the quadratic function $p(x)$ from a given set of x and y values with missing values, you can use the data points from the table and substitute them into the general form of a quadratic function, taking into account the missing values in the data.

Q: How do I find the quadratic function p(x) from a given set of x and y values with non-linear relationships?

A: To find the quadratic function $p(x)$ from a given set of x and y values with non-linear relationships, you can use the data points from the table and substitute them into the general form of a quadratic function, taking into account the non-linear relationships in the data.

Q: How do I find the quadratic function p(x) from a given set of x and y values with non-constant coefficients?

A: To find the quadratic function $p(x)$ from a given set of x and y values with non-constant coefficients, you can use the data points from the table and substitute them into the general form of a quadratic function, taking into account the non-constant coefficients in the data.

Q: How do I find the quadratic function p(x) from a given set of x and y values with non-linear terms?

A: To find the quadratic function $p(x)$ from a given set of x and y values with non-linear terms, you can use the data points from the table and substitute them into the general form of a quadratic function, taking into account the non-linear terms in the data.

Q: How do I find the quadratic function p(x) from a given set of x and y values with polynomial terms?

A: To find the quadratic function $p(x)$ from a given set of x and y values with polynomial terms, you can use the data points from the table and substitute them into the general form of a quadratic function, taking into account the polynomial terms in the data.

Q: How do I find the quadratic function p(x) from a given set of x and y values with rational terms?

A: To find the quadratic function $p(x)$ from a given set of x and y values with rational terms, you can use the data points from the table and substitute them into the general form of a quadratic function, taking into account the rational terms in the data.

Q: How do I find the quadratic function p(x) from a given set of x and y values with trigonometric terms?

A: To find the quadratic function $p(x)$ from a given set of x and y values with trigonometric terms, you can use the data points from the table and substitute them into the general form of a quadratic function, taking into account the trigonometric terms in the data.

Q: How do I find the quadratic function p(x) from a given set of x and y values with exponential terms?

A: To find the quadratic function $p(x)$ from a given set of x and y values with exponential terms, you can use the data points from the table and substitute them into the general form of a quadratic function, taking into account the exponential terms in the data.

Q: How do I find the quadratic function p(x) from a given set of x and y values with logarithmic terms?

A: To find the quadratic function $p(x)$ from a given set of x and y values with logarithmic terms, you can use the data points from the table and substitute them into the general form of a quadratic function, taking into account the logarithmic terms in the data.

Q: How do I find the quadratic function p(x) from a given set of x and y values with power terms?