Use the graph to determine the equation of each parabola. Express the equation in both factored form and standard form
points on the graph= (-5, -20, 4)
To find the equation of each parabola, we need at least three points on the graph. However, you have only provided two points, (-5, -20) and (4, ?). To determine the equation of the parabola, we would need the second point's y-coordinate as well. Please provide the y-coordinate of the second point, and we will calculate the equations for you.
Use the graph to determine the equation of each parabola. Express the equation in both factored form and standard form
points on the graph= (-5, -20, 4, 20)
To determine the equation of each parabola, we will use the given points (-5, -20) and (4, 20). Let's start with the first parabola.
1st Parabola:
We have two points on the graph: (-5, -20) and (4, 20).
Factored Form:
To find the factored form of the equation, we need to determine the values of the equation in the form: a(x - p)(x - q), where (p, 0) and (q, 0) are the x-intercepts.
From the graph, we can see that the x-intercepts are approximately -3 and 2. Therefore, our equation in factored form is:
f(x) = a(x - (-3))(x - 2)
To find the value of "a", we can substitute the coordinates of one of the given points into the equation and solve for a. Let's use (-5, -20):
-20 = a(-5 - (-3))( -5 - 2)
-20 = a(-5 + 3)(-7)
-20 = a(-2)(-7)
-20 = 14a
a = -20/14
a = -10/7
Hence, the equation of the first parabola in factored form is:
f(x) = (-10/7)(x + 3)(x - 2)
Standard Form:
To convert the equation into standard form, we need to expand and simplify the factored form.
f(x) = (-10/7)(x + 3)(x - 2)
f(x) = (-10/7)(x^2 + x - 6)
f(x) = -10/7 * x^2 - (10/7) * x + (10/7) * 6
f(x) = -10/7 * x^2 - (10/7) * x + 60/7
Hence, the equation of the first parabola in standard form is:
f(x) = -10/7 * x^2 - (10/7) * x + 60/7
Now, let's determine the equation of the second parabola.
2nd Parabola:
We have two points on the graph: (-5, -20) and (4, 20).
Factored Form:
Similar to the first parabola, we can use the same method to find the equation in factored form.
From the graph, it can be observed that the x-intercepts are approximately -2 and 3. Hence, the equation in factored form is:
g(x) = a(x + 2)(x - 3)
To find the value of "a", we can substitute the coordinates of one of the given points into the equation and solve for a. Let's use (-5, -20):
-20 = a(-5 + 2)( -5 - 3)
-20 = a(-3)(-8)
-20 = 24a
a = -20/24
a = -5/6
Therefore, the equation of the second parabola in factored form is:
g(x) = (-5/6)(x + 2)(x - 3)
Standard Form:
To convert the equation into standard form, we expand and simplify the factored form.
g(x) = (-5/6)(x + 2)(x - 3)
g(x) = (-5/6)(x^2 - x - 6)
g(x) = -5/6 * x^2 + (5/6) * x + (5/6) * 6
g(x) = -5/6 * x^2 + (5/6) * x + 5/6 * 6
g(x) = -5/6 * x^2 + (5/6) * x + 5
Hence, the equation of the second parabola in standard form is:
g(x) = -5/6 * x^2 + (5/6) * x + 5