A parabola has intercepts of x = -2, x = 3, and y = -4
how do I write this into these forms:
the intercept form (y = a (x – m)(x – n))
and
in standard (y = ax2 + bx + c) form
see
https://www.jiskha.com/display.cgi?id=1523149457
for the first part. I'm sure you can expand it and rearrange terms to get the standard form. If not, show what you come up with.
Well for the intercept form, I wrote -4 = a ( -2 – m) ( 3 – n) but I don't know which x to use for the standard form for a or b. or am i supposed to put the values of x's like this in standard form
(y = ax^2 + bx + c)
y = a -2^2 + b3 + c
To write the equation of a parabola in the intercept form (y = a (x - m)(x - n)), we need to find the values of a, m, and n using the given intercepts.
1. Intercepts x = -2 and x = 3:
When the x-intercepts are given, we know that the parabola crosses the x-axis at these points. The given intercepts are -2 and 3, which means that the roots of the equation are x = -2 and x = 3.
Therefore, (x + 2) and (x - 3) can be factors of the equation.
2. Intercept y = -4:
When the y-intercept is given, we know that the parabola crosses the y-axis at this point. The given y-intercept is -4, which means that when x = 0, y = -4.
Substituting the values into the equation:
y = a (x - m)(x - n)
-4 = a (0 - m)(0 - n)
-4 = a (-m)(-n)
-4 = a mn
Now, we have three equations:
1) x = -2: (x + 2) = 0
2) x = 3: (x - 3) = 0
3) y = -4: -4 = a mn
To find the equation, we need to solve for a, m, and n by either substituting or directly comparing the three equations.
From the first equation (x + 2) = 0, we get x = -2. Substituting this into equation 3 (-4 = a mn), we have:
-4 = a(-2)(n)
-4 = -2an
a = 2
From the second equation (x - 3) = 0, we get x = 3. Substituting this into equation 3 (-4 = a mn), we have:
-4 = a(m)(3)
-4 = 3am
a = -4/3m
Comparing the values of a from the two equations, we get:
2 = -4/3m
2 = -4/3m
6m = -4
m = -4/6
m = -2/3
Now that we have the values of a and m, we can substitute them back into equation 3 (-4 = a mn) to find n:
-4 = 2 (-2/3)(n)
-4 = -4n/3
-12 = -4n
n = 12/4
n = 3
Therefore, the equation of the parabola in intercept form (y = a (x - m)(x - n)) is:
y = 2 (x - (-2/3))(x - 3)
y = 2 (x + 2/3)(x - 3)
Now let's write the equation of the parabola in standard form (y = ax^2 + bx + c).
To convert the equation from intercept form to standard form, we need to expand and simplify:
y = 2 (x + 2/3)(x - 3)
y = 2 (x^2 - 3x + 2x - 6/3)
y = 2 (x^2 - x - 6/3)
y = 2 (x^2 - x - 2)
Expanding further:
y = 2x^2 - 2x - 4
Therefore, the equation of the parabola in standard form (y = ax^2 + bx + c) is:
y = 2x^2 - 2x - 4