Write the equation for the parabola that has x− intercepts (1.2,0) and (4,0) and y− intercept (0,12).
How about a real answer?
From the 2 x-intecepts we know it must be of the form
y = a(x-1.2)(x-4)
sub in the point (0,12)
12 = a(-1.2)(-4)
a = 2.5
y = 2.5(x - 1.2)(x - 4)
expand and simplify if you feel it necessary
To find the equation of a parabola given its x-intercepts and y-intercept, we can use the vertex form of the equation, which is in the form of:
y = a(x - h)^2 + k
where (h, k) represents the vertex of the parabola.
Step 1: Finding the vertex:
Since the x-intercepts are (1.2, 0) and (4, 0), the average of the x-coordinates will give us the x-coordinate of the vertex.
x-coordinate of the vertex = (1.2 + 4) / 2 = 5.2 / 2 = 2.6
Since the y-intercept is (0, 12), the y-coordinate of the vertex will be 12.
Therefore, the vertex is (2.6, 12).
Step 2: Finding the value of a:
Using the vertex form, we can substitute the coordinates of the vertex and the y-intercept into the equation, and solve for a.
12 = a(0 - 2.6)^2 + 12
0 = 6.76a
a = 0
Step 3: Writing the equation:
Now that we have the value of a as 0, the equation of the parabola becomes:
y = 0(x - 2.6)^2 + 12
y = 12
Therefore, the equation of the parabola is y = 12.
To write the equation for a parabola, we need to use the standard form of the quadratic equation, which is y = ax^2 + bx + c.
Given that the parabola has x-intercepts at (1.2, 0) and (4, 0), we can infer that the parabola crosses the x-axis at those points. In other words, when x = 1.2 and x = 4, y = 0.
Substituting these values into the equation, we get two equations:
0 = a(1.2)^2 + b(1.2) + c ... (Equation 1)
0 = a(4)^2 + b(4) + c ... (Equation 2)
Next, we are given that the y-intercept is at the point (0, 12). By substituting x = 0 and y = 12 into the standard form equation, we can find another equation:
12 = a(0)^2 + b(0) + c
12 = c ... (Equation 3)
Now we have three equations: Equation 1, Equation 2, and Equation 3. We can solve this system of equations simultaneously to find the values of a, b, and c, which will allow us to write the equation for the parabola.
Substituting Equation 3 into Equation 1 and Equation 2, we get:
0 = a(1.2)^2 + b(1.2) + 12
0 = a(4)^2 + b(4) + 12
Simplifying these equations, we have:
1.44a + 1.2b + 12 = 0 ... (Equation 4)
16a + 4b + 12 = 0 ... (Equation 5)
Now we have a system of two equations (Equation 4 and Equation 5) with two variables (a and b). Solving this system of equations will give us the values of a and b.
Solving Equation 4 for a, we get:
a = (-1.2b - 12) / 1.44
Substituting this value of a into Equation 5, we have:
16((-1.2b - 12) / 1.44) + 4b + 12 = 0
Simplifying this equation, we find:
-10.78b - 12 = 0
Solving for b, we get:
b = -12 / -10.78
b ≈ 1.11
Substituting this value of b into Equation 4, we can solve for a:
1.44a + 1.2(1.11) + 12 = 0
Simplifying this equation, we find:
1.44a + 1.33 + 12 = 0
1.44a = -13.33
a ≈ -9.25
Now that we have the values of a, b, and c, we can write the equation for the parabola:
y = -9.25x^2 + 1.11x + 12