Question: Write the equation of a cubic function that has the x-intercepts of -4, 2, and 5 and passes through the point (6,20)
My Answer: y = .5x^3-1.5x^2-9x+20
What I did: Put the x-intercepts and given point into stat plot (on calculator) then hit stat -> calc -> cubic reg on calculator and was given that equation however it doesn't pass through the given point (6,20)
Can you help and tell me if I did anything wrong?
Thank You
I will do mine using just a good ol' fashioned pencil and paper ...
intercepts are -4, 2, and 5
so
y = a(x+4)(x-2)(x-5)
The a will stretch it and force it to go through (6,20)
20 = a(10)(4)(1)
20 = 40a
a = 1/2
y = (1/2)(x+4)(x-2)(x-5)
Well, what do you know, I didn't even need my paper and pencil, just simple mental arithmetic
I see thanks for the help but what is the final equation? I had 1/2x^3-x.5x^2-9x+20
What is wrong with my "final" equation.
Why mess up a nice looking equation in factored form ??
your answer is correct, if you fix that typo in "-x.5x^2"
5/3,-5,-1 and passes through the point (0,50)
To find the equation of a cubic function that passes through the given x-intercepts and point, you need to use the Interpolation method rather than using the calculator's regression analysis.
Here's how you can approach it step by step:
Step 1: Start with the general form of a cubic function: y = ax^3 + bx^2 + cx + d.
Step 2: Use the x-intercepts to determine the factors of the equation. Since the x-intercepts are -4, 2, and 5, this means that the factors will be (x + 4), (x - 2), and (x - 5). So the equation can be written as:
y = a(x + 4)(x - 2)(x - 5).
Step 3: Now, we need to determine the value of 'a'. To do this, we can substitute the given point (6, 20) into the equation.
20 = a(6 + 4)(6 - 2)(6 - 5).
Simplifying:
20 = a(10)(4)(1).
20 = 40a.
Dividing both sides by 40, we find:
a = 20/40 = 1/2.
Step 4: Substitute the value of 'a' back into the equation:
y = (1/2)(x + 4)(x - 2)(x - 5).
Simplify further if needed.
Therefore, the correct equation for the cubic function that passes through the x-intercepts -4, 2, and 5 and the given point (6, 20) is:
y = (1/2)(x + 4)(x - 2)(x - 5).