### Writing an Equation for a Polynomial FunctionIn this exercise, you are given a graph of a polynomial function and asked to determine its algebraic equation.#### Graph AnalysisThe graph presented has the following key characteristics:- It appears to be a smooth curve, which is typical of polynomial functions.- The function has three roots (where it crosses the x-axis) at approximately $ x = -3 $, $ x = 1$, and $ x = 3 $.- The curve changes direction at four points, indicating turning points or local extrema.- The polynomial appears to be of degree 4, as it has three turning points.#### Identifying the Polynomial FunctionConsidering the general form of a polynomial equation:\[ f(x) = a(x + 3)(x - 1)(x - 3) \]where $ a $ is a scalar multiplier that defines the vertical stretch or shrink of the function.#### Graph Point Analysis- **X-intercepts**: The polynomial has roots at $ x = -3 $, $ x = 1 $, and $ x = 3 $.- **Y-intercept**: The value of the function when $ x = 0 $. Visually locating the point of intersection on the y-axis helps to determine the coefficient $ a $.### Equation ConstructionGiven these roots, the polynomial function in factored form could be expressed as:\[ f(x) = a(x + 3)(x - 1)(x - 3) \]To determine the exact polynomial, the value of $ a $ can be calculated by substituting a specific point on the graph into the function.### ExampleSuppose the graph passes through the point $(0, -6)$:\[ -6 = a(0 + 3)(0 - 1)(0 - 3) \]\[ -6 = a(3)(-1)(-3) \]\[ -6 = 9a \]\[ a = -\frac{6}{9} = -\frac{2}{3} \]Thus, the polynomial function is approximately:\[ f(x) = -\frac{2}{3}(x + 3)(x - 1)(x - 3) \]### Final Polynomial Function\[ f(x) = -\frac{2}{3} (x + 3)(x - 1)(x -

We have to find the equation for the given polynomial function.

Answered: Write an equation for the polynomial…

Math

Calculus

Write an equation for the polynomial function graphed. -3 -2 -1 -2 -4 f(x) = 67

College Algebra

7th Edition

ISBN:9781305115545

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:James Stewart, Lothar Redlin, Saleem Watson

Chapter3: Polynomial And Rational Functions

Section: Chapter Questions

Problem 1P

See similar textbooks

Related questions

Q: Write an equation for the polynomial graphed below. +'? -4 f(x) = 8 7 6- 5 4 3 2 ņ m A -5- OA -6-…

A: Write the equation of the polynomial

Q: Write an equation for the polynomial graphed below 5 -5-4-3-2 y(x) = 1 -2 -3 -4 -5 1 2 3 4 5 Q

A: We have to construct the polynomial for the given graph.

Q: 40),(3

Q: Determine the equation for the polynomial function that is represented by the given graph. Please…

Q: Write an equation for the polynomial graphed below -5 -4 -3 -2 y(x) = 7 6+ 5+ 4+ 3 + 2+ 1+ -1 -2- -3…

Q: Write an equation for the polynomial graphed below is t 13 -2 -1 AM 2 7 123456 -1 -2+ -3 -4 7 2 3 4…

Q: To find the distance across a small lake, a surveyor has taken the measurements shown. Find the…

A: As per guideline we can answer only one question, Please repost other question.

Q: Write an equation for the polynomial function whose graph is shown: -5 4-3 1 2 34 -2

A: Roots of a polynomial refer to the values of a variable for which the given polynomial is equal to…

Q: Write an equation for the polynomial graphed below -5 1-4 -3 -2 -1 6 5 4 3 2 1 -2 -3 -4 -5 -6+ 2 3 4…

Concept explainers

Roots

The main dissimilarity between real and complex roots is that the real roots are obtained as real numbers, in contrast with the complex roots, which are indicated as imaginary numbers. The standard format for depicting any complex number is in the form of a + bi.

Question

### Writing an Equation for a Polynomial Function

In this exercise, you are given a graph of a polynomial function and asked to determine its algebraic equation.

#### Graph Analysis

The graph presented has the following key characteristics:

- It appears to be a smooth curve, which is typical of polynomial functions.
- The function has three roots (where it crosses the x-axis) at approximately $ x = -3 $, $ x = 1$, and $ x = 3 $.
- The curve changes direction at four points, indicating turning points or local extrema.
- The polynomial appears to be of degree 4, as it has three turning points.

#### Identifying the Polynomial Function

Considering the general form of a polynomial equation:
\[ f(x) = a(x + 3)(x - 1)(x - 3) \]
where $ a $ is a scalar multiplier that defines the vertical stretch or shrink of the function.

#### Graph Point Analysis

- **X-intercepts**: The polynomial has roots at $ x = -3 $, $ x = 1 $, and $ x = 3 $.
- **Y-intercept**: The value of the function when $ x = 0 $. Visually locating the point of intersection on the y-axis helps to determine the coefficient $ a $.

### Equation Construction

Given these roots, the polynomial function in factored form could be expressed as:
\[ f(x) = a(x + 3)(x - 1)(x - 3) \]

To determine the exact polynomial, the value of $ a $ can be calculated by substituting a specific point on the graph into the function.

### Example

Suppose the graph passes through the point $(0, -6)$:
\[ -6 = a(0 + 3)(0 - 1)(0 - 3) \]
\[ -6 = a(3)(-1)(-3) \]
\[ -6 = 9a \]
\[ a = -\frac{6}{9} = -\frac{2}{3} \]

Thus, the polynomial function is approximately:
\[ f(x) = -\frac{2}{3}(x + 3)(x - 1)(x - 3) \]

### Final Polynomial Function

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

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Step by step

Solved in 2 steps with 1 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Roots$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

Similar questions

Calculus The graph of a cubic polynomial function has horizontal tangent lines at (1,2) and (1,2). Find an equation for the function and sketch its graph.
Height of a basketball The path of a basketball thrown from the free throw line can be modeled by the quadratic function f(x)=0.06x2+1.5x+6, where x is the horizontal distance in feet from the free throw line and f(x) is the height in feet of the ball. Find the maximum height of the basketball.
Writing an Equation The graph of a fourth-degree polynomial function y=fx is shown. The equation has 2i as zeros. Write an equation for f.
An outpatient operating room charges each patient a fixed amount per surgery plus an amount per minute for use. One patient was charged 250 for a 30-minute surgery and another patient was charged 450 for a 90-minute surgery. Determine the linear function that would be used to compute the charges. Use the function to determine the charge when a patient has a 45-minute surgery.
Practice Write a third-degree polynomial function with real coefficients and the given zeros. 1,i
Fill in the blanks. A function is a second-degree polynomial function, and its graph is called a .
Use the descriptions to write an equation of a polynomial function with the given characteristics, "Touches the x-axis at 0 and crosses the x-axis at 2; lies below the x-axis between 0 and 2". Use a graphing utility to graph your function to see if you are correct. If not, modify the function’s equation and repeat this process.
What is meant by the degree of a polynomial function? Show an example of how we can classify a polynomial by degree. What are the four possible scenarios of end behavior for a polynomial function? Describe them and give an example of each. How is end behavior related to slope? Will a polynomial function always have the same number of zeros as its degree? How does this relate to multiplicity?
Explain the steps to sketch a graph of this factored form polynomial function. f(x)= (x-2)(x-1)3
F(x)=(x-1)^2-2 Sketch the graph of each function.use the axis of symmetry ,Y intercept ,X intercepts or a table of values Also show the work thank you
Using the attached graph, sketch the derivative. Graph of polynomial f(x) -3 -3 -4 -5 -6
Consider the function f(x) = x2 + 3x - 4. (a) Factor this polynomial and also complete the square. Draw a graph of the function. Label the roots and vertex. (b) This function is not linear, so the "slope" changes as we change the x value. We will investigate that here. Find the slope of the line that connects the points (1, f(1)) and (4, f(4)) on the graph of f. (c) We define a new function gı (2) which finds the slope of the line between the points (1, f(1)) and (x, f(x)) on the graph of f. In the last step you calculated 91(4). Find 91(2), 91(1.5), 91(1.1), and 91(1.01). (d) Explain why (1) is undefined. Based on the values you calculated in part (c). what do you think gı(1) should be? Use this value for 91(1) from now on. (e) Find the equation of the line through (1, f(1)) with slope 91 (2) and draw it along with the graph of f. What do you notice about how the line relates to the parabola? (f) Define 93 (2) to be the slope of the line through (3, f(3)) and (x, f(x)). Again 93(3) is…