Sketch The Graph Of A Polynomial With Zeros At $-3$, $2$ (with A Multiplicity Of 2), And $5$, And A $y$-intercept At $-4$.

Feb 26, 2025 by ADMIN 123 views

**Sketching the Graph of a Polynomial with Given Zeros and Y-Intercept**

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Introduction

In this article, we will explore the process of sketching the graph of a polynomial function with given zeros and a y-intercept. The zeros of a polynomial function are the values of x that make the function equal to zero, while the y-intercept is the value of y when x is equal to zero. Understanding how to sketch the graph of a polynomial function with given zeros and a y-intercept is an essential skill in algebra and mathematics.

Understanding the Problem

The problem asks us to sketch the graph of a polynomial function with zeros at $-3$ , $2$ (with a multiplicity of 2), and $5$ , and a y-intercept at $-4$ . This means that the polynomial function will have factors of $(x+3)$ , $(x-2)^2$ , and $(x-5)$ , and the constant term will be $-4$ .

Writing the Polynomial Function

To write the polynomial function, we need to multiply the factors together and set the constant term equal to $-4$ . The polynomial function can be written as:

f(x) = a(x+3)(x-2)^2(x-5)

where $a$ is a constant that we need to determine.

Determining the Constant a

To determine the constant $a$ , we can use the fact that the y-intercept is $-4$ . When $x=0$ , the value of $f(x)$ is equal to $-4$ . Substituting $x=0$ into the polynomial function, we get:

f(0) = a(0+3)(0-2)^2(0-5) = -4

Simplifying the expression, we get:

-12a = -4

Dividing both sides by $-12$ , we get:

a = \frac{1}{3}

Writing the Final Polynomial Function

Now that we have determined the constant $a$ , we can write the final polynomial function:

f(x) = \frac{1}{3}(x+3)(x-2)^2(x-5)

Sketching the Graph

To sketch the graph of the polynomial function, we need to identify the zeros and the y-intercept. The zeros are the values of x that make the function equal to zero, and the y-intercept is the value of y when x is equal to zero.

The zeros of the polynomial function are $-3$ , $2$ (with a multiplicity of 2), and $5$ . This means that the graph of the polynomial function will have vertical asymptotes at $x=-3$ , $x=2$ , and $x=5$ .

The y-intercept of the polynomial function is $-4$ . This means that the graph of the polynomial function will pass through the point $(0,-4)$ .

Graphing the Polynomial Function

To graph the polynomial function, we can use the following steps:

Plot the zeros: Plot the points $(-3,0)$ , $(2,0)$ , and $(5,0)$ on the graph.
Plot the y-intercept: Plot the point $(0,-4)$ on the graph.
Draw the graph: Draw a smooth curve through the points, making sure to include the vertical asymptotes at $x=-3$ , $x=2$ , and $x=5$ .

Conclusion

In this article, we have explored the process of sketching the graph of a polynomial function with given zeros and a y-intercept. We have written the polynomial function, determined the constant $a$ , and sketched the graph of the polynomial function. The graph of the polynomial function has vertical asymptotes at $x=-3$ , $x=2$ , and $x=5$ , and passes through the point $(0,-4)$ .

Final Answer

The final answer is the graph of the polynomial function:

f(x) = \frac{1}{3}(x+3)(x-2)^2(x-5)

This graph has vertical asymptotes at $x=-3$ , $x=2$ , and $x=5$ , and passes through the point $(0,-4)$ .

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Introduction

In our previous article, we explored the process of sketching the graph of a polynomial function with given zeros and a y-intercept. We wrote the polynomial function, determined the constant $a$ , and sketched the graph of the polynomial function. In this article, we will answer some frequently asked questions about sketching the graph of a polynomial function with given zeros and a y-intercept.

Q: What are the zeros of a polynomial function?

A: The zeros of a polynomial function are the values of x that make the function equal to zero. In other words, they are the values of x that satisfy the equation $f(x) = 0$ .

Q: How do I find the zeros of a polynomial function?

A: To find the zeros of a polynomial function, you can use the factoring method or the quadratic formula. If the polynomial function can be factored, you can set each factor equal to zero and solve for x. If the polynomial function cannot be factored, you can use the quadratic formula to find the zeros.

Q: What is the y-intercept of a polynomial function?

A: The y-intercept of a polynomial function is the value of y when x is equal to zero. In other words, it is the point on the graph where the graph intersects the y-axis.

Q: How do I find the y-intercept of a polynomial function?

A: To find the y-intercept of a polynomial function, you can substitute x = 0 into the polynomial function and solve for y.

Q: What is the difference between a zero and a root of a polynomial function?

A: A zero of a polynomial function is a value of x that makes the function equal to zero, while a root of a polynomial function is a value of x that makes the function equal to zero and is also a solution to the equation $f(x) = 0$ . In other words, all zeros are roots, but not all roots are zeros.

Q: How do I sketch the graph of a polynomial function with given zeros and a y-intercept?

A: To sketch the graph of a polynomial function with given zeros and a y-intercept, you can follow these steps:

Plot the zeros: Plot the points where the graph intersects the x-axis.
Plot the y-intercept: Plot the point where the graph intersects the y-axis.
Draw the graph: Draw a smooth curve through the points, making sure to include the vertical asymptotes at the zeros.

Q: What are some common mistakes to avoid when sketching the graph of a polynomial function with given zeros and a y-intercept?

A: Some common mistakes to avoid when sketching the graph of a polynomial function with given zeros and a y-intercept include:

Not plotting the zeros: Make sure to plot the points where the graph intersects the x-axis.
Not plotting the y-intercept: Make sure to plot the point where the graph intersects the y-axis.
Not including the vertical asymptotes: Make sure to include the vertical asymptotes at the zeros.
Not drawing a smooth curve: Make sure to draw a smooth curve through the points.

Conclusion

In this article, we have answered some frequently asked questions about sketching the graph of a polynomial function with given zeros and a y-intercept. We have discussed the zeros and roots of a polynomial function, the y-intercept, and how to sketch the graph of a polynomial function with given zeros and a y-intercept. We have also discussed some common mistakes to avoid when sketching the graph of a polynomial function with given zeros and a y-intercept.

Final Answer

The final answer is that sketching the graph of a polynomial function with given zeros and a y-intercept requires a clear understanding of the zeros and roots of a polynomial function, the y-intercept, and how to plot the points and draw a smooth curve through the points.