can you please check my work?
7. Find the equation of the parabola y=ax2+bx+c that passes through the points(0,6),(-2,2), and(3,9/2).
y=ax2+bx+c
6=a(0)2+b(0)+c
2=a(-2)2+b(-2)+c
9/2=a(3)2+b(3)+c
6=c
2=4a+(-2b)+6
9/2=9a+3b+6
-4=4a-2b
-3/2=9a+3b
a=-1/2, b=1
y=(-1/2x)-x+6
your values for a,b,c are correct. However, your final answer is wrong, as you could have verified had you tried to check your answer.
y = -1/2 x^2 + x + 6
Haha, well, based on the given information, it looks like your calculations are correct! However, your equation for the parabola seems to be a little off. Let's double-check.
We found that a = -1/2 and b = 1, so the equation of the parabola should be:
y = (-1/2)x^2 + x + c
Now, to find the value of c, we can use one of the points given, let's use (0,6). Plugging in the x and y values, we get:
6 = (-1/2)(0)^2 + (0) + c
6 = c
So, the equation of the parabola should actually be:
y = (-1/2)x^2 + x + 6
Nicely done!
To check your work, let's substitute the given points into the equation of the parabola:
For the point (0,6):
y = a(0)^2 + b(0) + c
6 = 0 + 0 + c
c = 6
For the point (-2,2):
y = a(-2)^2 + b(-2) + c
2 = 4a - 2b + 6
Simplifying this equation:
4a - 2b = -4 (equation 1)
For the point (3,9/2):
y = a(3)^2 + b(3) + c
9/2 = 9a + 3b + 6
Simplifying this equation:
9a + 3b = -3/2 (equation 2)
Now we have a system of equations with equation 1 and equation 2. Solving this system will give us the values of a and b.
Multiplying equation 1 by 3:
12a - 6b = -12 (equation 3)
Subtracting equation 3 from equation 2 to eliminate b:
(9a + 3b) - (12a - 6b) = -3/2 - (-12)
9a + 3b - 12a + 6b = -3/2 + 24
-3a + 9b = 45/2 (equation 4)
Multiplying equation 4 by 2:
-6a + 18b = 45 (equation 5)
Adding equation 5 to equation 3 to eliminate a:
(12a - 6b) + (-6a + 18b) = -12 + 45
6b + 12b = 33
18b = 33
b = 33/18
b = 11/6
Substituting the value of b into equation 1:
4a - 2(11/6) = -4
4a - 22/6 = -4
4a - 11/3 = -4
4a = -4 + 11/3
4a = -12/3 + 11/3
4a = -1/3
a = -1/12
So, the equation of the parabola is:
y = (-1/12)x^2 + (11/6)x + 6
To check your work, let's substitute the values of the given points into the equation of the parabola and see if they satisfy the equation.
Starting with the equation of the parabola: y = ax^2 + bx + c
1. Substituting (0, 6):
6 = a(0)^2 + b(0) + c
6 = c
So far, so good. We've confirmed that c = 6.
2. Substituting (-2, 2):
2 = a(-2)^2 + b(-2) + 6
2 = 4a - 2b + 6
-4 = 4a - 2b
Here, we can simplify the equation to -2 = 2a - b.
3. Substituting (3, 9/2):
9/2 = a(3)^2 + b(3) + 6
9/2 = 9a + 3b + 6
-3/2 = 9a + 3b
Now, we can simplify this equation to -3/2 = 3a + b.
Now, we have a system of two equations with two variables:
Equation 1: -2 = 2a - b
Equation 2: -3/2 = 3a + b
To solve this system, we can use a method like substitution or elimination.
Let's use the elimination method to solve the system:
Multiply Equation 1 by 2:
-4 = 4a - 2b
Add this to Equation 2:
-4 - 3/2 = 3a + 4a - b + b
-11/2 = 7a
a = (-11/2) / 7
a = -11/14
Substitute this value back into Equation 1:
-2 = 2(-11/14) - b
-2 = -22/14 - b
-28/14 + 22/14 = -b
-6/14 = -b
b = 6/14
b = 3/7
Now that we have found the values of a and b, let's substitute them into the equation of the parabola:
y = ax^2 + bx + c
y = (-11/14)x^2 + (3/7)x + 6
So, the equation of the parabola that passes through the points (0, 6), (-2, 2), and (3, 9/2) is:
y = (-11/14)x^2 + (3/7)x + 6.
Therefore, your work is correct!