Sketch the graph of a polynomial of degree 3 with a positive leading coefficient whose zeros are -4, -2 and 3
Follow the same steps as in the post when you were Lay.
To sketch the graph of a polynomial of degree 3 with specific zeros, we need to follow a few steps:
Step 1: Identify the zeros.
The given zeros are -4, -2, and 3.
Step 2: Determine the leading coefficient.
The problem states that the leading coefficient is positive. Let's assume it is 1 for simplicity.
Step 3: Write the polynomial equation.
In general form, for a polynomial of degree 3 with leading coefficient "a" and zeros "p", "q", and "r", the equation is:
f(x) = a(x - p)(x - q)(x - r)
Plugging in the given zeros, we have:
f(x) = (x + 4)(x + 2)(x - 3)
Step 4: Simplify the equation.
Expand and simplify the equation to get the polynomial in its standard form:
f(x) = (x^2 + 6x + 8)(x - 3)
f(x) = x^3 + 6x^2 + 8x - 3x^2 - 18x - 24
f(x) = x^3 + 3x^2 - 10x - 24
Step 5: Sketch the graph.
Now that we have the equation f(x) = x^3 + 3x^2 - 10x - 24, we can plot the graph. The y-coordinate at any point on the x-axis represents f(x).
By analyzing the coefficient signs, we can determine the behavior of the function between the zeros:
- The cubic term (x^3) indicates the graph will be either upward or downward.
- The squared term (3x^2) with a positive coefficient suggests the graph will be upward.
- The linear term (-10x) with a negative coefficient shows the graph will be downward.
- The constant term (-24) gives us the y-intercept.
Based on these observations, we know that the graph will start in the negative y-region and end in the positive y-region because of the positive leading coefficient.
By calculating f(0), we find that f(0) = -24. This means that the graph intersects the y-axis at y = -24.
Using the zeros we were given (-4, -2, and 3), we can plot these points on the x-axis as well. Thus, the graph will pass through (-4, 0), (-2, 0), and (3, 0).
Now, by connecting these points and considering the behavior between the zeros indicated by the coefficient signs, we can sketch the graph of the polynomial. Remember, the exact shape will depend on the scale and size chosen for the graph.
The resulting graph should show a curve that starts in the third quadrant, crossing the x-axis at -4, then going through the x-axis at -2 and 3, and finally ending in the first quadrant. The graph will also intersect the y-axis at y = -24.