Can you help me with the following questions, I don't understand. thanks!
Write the equation of a polynomial function with the given characteristics. Use a leading coefficient of 1 or -1 and make the degree of the function as small as possible.
#1. Crosses the x-axis at -4, 0, and 1; lies above the x-axis between -4 and 0; lies below the x-axis between 0 and 1.
#2.Touches the x-axis at 0 and crosses the x-axis at 4; lies below the x-axis between 0 and 4
for #1
x-intercepts of a function are solutions to the corresponding equation
since there are solutions of -4,0,and 1
the function must have been f(x) = (x+4)x-0)(x-1)
or
f(x) = x(x-1)(x+4) . You can expand it or leave it as is
for #2
if a graph "touches" the x-axis then there must have been a double root, such as (x-2)(x-2)
so this one clearly must have been
f(x) = (x-0)(x-0)(x-4)
= x^2(x-4)
In both cases we need a cubic for 3 roots. If the x^3 is positive, then the cubic "rises" into the first quadrant and "drops" into the third quadrant.
Since this was the case for both functions there would be a +1 understood in front of the factors,
had it been reversed, a -1 in front of the factors would have achieved this
Sure, I can help you with that! Let's break down the questions step by step.
#1. To write the equation of a polynomial function with the given characteristics, start by using the x-intercepts. This means the function will have factors of (x -(-4)), (x - 0), and (x - 1) since it crosses the x-axis at -4, 0, and 1 respectively.
Next, consider the behavior of the graph between the x-intercepts. Since it lies above the x-axis between -4 and 0 and lies below the x-axis between 0 and 1, we can assume that the degree of the polynomial must be odd in order to exhibit these characteristics.
The smallest possible degree is 3, which means the polynomial will be a cubic function. Since we want to use a leading coefficient of 1 or -1, let's choose a leading coefficient of 1.
Putting it all together, the equation of the polynomial function is:
f(x) = (x + 4)(x)(x - 1)
#2. Similar to the first question, start by using the x-intercepts. The function touches the x-axis at 0, so we know it has a factor of (x - 0) = x. It also crosses the x-axis at 4, so we know it has a factor of (x - 4).
Now, consider the behavior of the graph between the x-intercepts. Since it lies below the x-axis between 0 and 4, we can assume that the degree of the polynomial must be odd to exhibit this characteristic.
The smallest possible degree is 1, which means the polynomial will be a linear function. Since we want to use a leading coefficient of 1 or -1, let's choose a leading coefficient of -1.
Putting it all together, the equation of the polynomial function is:
f(x) = -x(x - 4)
I hope this helps! Let me know if you have any more questions.
Of course, I'd be happy to help you with these questions!
For question #1, we're given that the polynomial function crosses the x-axis at -4, 0, and 1. This means that these three values will be the roots or zeroes of the polynomial. Since the function lies above the x-axis between -4 and 0, and below the x-axis between 0 and 1, this suggests that the function changes sign at each of these x-values.
To find the equation of this polynomial function, we'll first identify its factors. Since the function crosses the x-axis at -4, 0, and 1, these will be the roots, and we can write the corresponding factors as (x + 4), x, and (x - 1), respectively.
Since the degree of the function should be as small as possible, we can combine these factors into a quadratic polynomial by multiplying them together:
f(x) = (x + 4)(x)(x - 1)
This simplifies to:
f(x) = x^3 + 3x^2 - 4x
So, the equation of the polynomial function with these characteristics is f(x) = x^3 + 3x^2 - 4x.
For question #2, we're told that the polynomial function touches the x-axis at 0 and crosses the x-axis at 4. This means that 0 will be a root of the polynomial, and 4 will be another root.
Similarly to question #1, we'll start by identifying the factors. The factor corresponding to the root at 0 is simply x, while the factor corresponding to the root at 4 is (x - 4).
Since we want the smallest degree for the function, we can multiply these factors together to form a linear polynomial:
f(x) = x(x - 4)
This simplifies to:
f(x) = x^2 - 4x
So, the equation of the polynomial function with these characteristics is f(x) = x^2 - 4x.
Remember to double-check your answers and make sure they satisfy all the given characteristics. I hope this helps! Let me know if you have any further questions.