x -4 -3 -2 -1 0 1 2 3 4
y 18 8 0 -6 -10 -12 -12 -10 -6
What is the vertex of the parabola? The x-coordinate is 1.5.
I can't read you numbers, but the y vertex is between the two -12 -12, probably half way. If that occurs when x=1.5, correct.
The numbers are
x -4 -3 -2 -1 0 1 2 3 4
y 18 8 0 -6 -10 -12 -12 -10 -6
To find the vertex of a parabola, we need to use the standard form of a quadratic equation: y = ax^2 + bx + c.
We have a set of points (x, y) given. Let's focus on the three points closest to the vertex, which are (-2, -6), (-1, -10), and (0, -12). By observing the y-values, we can see that the y-values decrease as we move towards the vertex and then increase as we move away from it. This pattern indicates that the vertex lies in between the two points (-2, -6) and (-1, -10).
To find the x-coordinate of the vertex, we can use the formula x = (-b) / (2a), where a and b are the coefficients of the quadratic equation.
Let's rearrange the equation y = ax^2 + bx + c to the form ax^2 + bx + c - y = 0.
By substituting the values of the point P(-2, -6) into the equation, we get:
a(-2)^2 + b(-2) + c - (-6) = 0
4a - 2b + c + 6 = 0
Similarly, substituting the values of the point Q(-1, -10) into the equation, we get:
a(-1)^2 + b(-1) + c - (-10) = 0
a - b + c + 10 = 0
We now have a system of equations:
4a - 2b + c = -6 (Equation 1)
a - b + c = -10 (Equation 2)
To solve this system of equations, we can subtract Equation 2 from Equation 1 to eliminate the variable c:
3a - b = 4 (Equation 3)
Since we now have a system of two equations with two unknowns (Equations 2 and 3), we can solve for a and b. Let's solve Equation 3 for b:
b = 3a - 4 (Equation 4)
Substituting Equation 4 into Equation 2, we get:
a - (3a - 4) + c = -10
a - 3a + 4 + c = -10
-2a + 4 + c = -10
-2a + c = -14 (Equation 5)
Now, we can substitute Equation 4 and Equation 5 into Equation 1 to solve for a and c:
4a - 2(3a - 4) + c = -6
4a - 6a + 8 + c = -6
-2a + 8 + c = -6
-2a + c = -14 (Equation 6)
Comparing Equation 5 and Equation 6, we can see they are the same equation, meaning that a = -2 and c = -14.
Substituting a = -2 into Equation 4, we can solve for b:
b = 3(-2) - 4
b = -6 - 4
b = -10
Therefore, we have the values a = -2, b = -10, and c = -14.
The equation of the parabola is y = -2x^2 - 10x - 14.
To find the x-coordinate of the vertex, we use the formula x = (-b) / (2a):
x = (-(-10)) / (2(-2))
x = 10 / -4
x = -2.5
Since the given x-coordinate of the vertex is 1.5, we can conclude that the given information does not match the equation of the parabola.