Sketch the graph of a 3rd degree polynomial function with a positive leading coefficient and 3 real zeros at x = - 2 , x = 1 , x = 3
up from lower left, cross at the zeroes, and proceed up to the right.
You know what cubic graphs look like, right?
To sketch the graph of a 3rd degree polynomial function with three real zeros at x = -2, x = 1, and x = 3, we first need to determine the equation of the polynomial.
Since we are given three real zeros, we can write the polynomial in factored form using the zero-product property:
f(x) = a(x - r1)(x - r2)(x - r3)
where a is the leading coefficient and r1, r2, and r3 are the zeros.
Given that the leading coefficient is positive, let's assume a = 1 for simplicity. So the equation becomes:
f(x) = (x + 2)(x - 1)(x - 3)
Now we can expand this equation:
f(x) = (x + 2)(x - 1)(x - 3)
= (x^2 + x - 2)(x - 3)
= x^3 - 3x^2 + x^2 - 3x - 2x + 6
= x^3 - 2x^2 - 5x + 6
We have the equation of the polynomial, so now let's graph it.
The graph of a polynomial with degree 3 can take different shapes, but due to the given information, we can conclude the following:
- The leading coefficient is positive, which means the graph will start in the lower left quadrant and end in the upper right quadrant.
- The zeros are at x = -2, x = 1, and x = 3. These are three distinct real zeros, which means the graph will cross the x-axis at these points.
- Since the degree of the polynomial is odd, there will be no symmetry about the y-axis.
Plotting these points on the graph and considering the shape of the polynomial, we can sketch the graph as follows:
^
|
|
+---+----+---+---+----+----->
-2 1 3
Please note that this is a rough sketch of the graph, and you may want to use a graphing tool or specific software to get a more accurate representation.
To sketch the graph of a 3rd degree polynomial function with the given conditions, follow these steps:
1. Determine the equation of the polynomial function:
Since the polynomial has 3 real zeros at x = -2, x = 1, and x = 3, we know that the factors of the polynomial are (x + 2), (x - 1), and (x - 3).
To find the equation, we can multiply these factors together using the distributive property:
f(x) = (x + 2)(x - 1)(x - 3)
2. Determine the leading coefficient:
The question states that the leading coefficient is positive. However, since the polynomial is given in factored form, we know that the leading coefficient will be positive if the highest power of x (degree) is even, and negative if the degree is odd.
Since the degree of the polynomial is 3, an odd number, multiply the polynomial by a positive constant to make the leading coefficient positive:
f(x) = a(x + 2)(x - 1)(x - 3) (where 'a' is a positive constant)
3. Find 'a' by considering the behavior at each zero:
To find the constant 'a' and complete the equation, we consider the behavior of the graph at each zero.
At x = -2, we know that the function will cross the x-axis, so 'a' should be positive.
At x = 1, we know that the function will bounce off the x-axis (change direction), so 'a' should be negative.
At x = 3, we know that the function will cross the x-axis, so 'a' should be positive.
Based on this analysis, let's assume that 'a' = 1, which means the equation is:
f(x) = (x + 2)(x - 1)(x - 3)
4. Determine the behavior of the graph as x approaches positive and negative infinity:
Consider the term with the highest exponent in the equation. In this case, we have x^3, which indicates that as x approaches both positive and negative infinity, the graph will also approach positive and negative infinity.
5. Plot the x-intercepts:
The zeros of the polynomial are already given as x = -2, x = 1, and x = 3. Plot these points on the x-axis.
6. Determine the end behavior and the overall shape of the graph:
Since the leading coefficient is positive and the degree is odd, the graph will have opposite behavior from the leftmost end to the rightmost end. This means the graph will start at the bottom left, go through the x-intercept at x = -2, bounce off the x-intercept at x = 1, go through the x-intercept at x = 3, and extend towards the top right as x approaches positive infinity.
7. Sketch the curve by connecting the dots:
Connect the plotted points using a smooth curve based on the behavior you determined in step 6. Make sure the curve goes through the x-intercepts and follows the general shape determined.
Following these steps, you will have a sketch of the graph of the 3rd degree polynomial function with a positive leading coefficient and 3 real zeros at x = -2, x = 1, and x = 3.