I started having trouble on part (c), and I need to know this before
I can go onto the other problems. Plz help!
The parabola shown at the right (no picture, sorry) is of the form y=x^2+bx+c. The graph's y-intercept is (0,2), the axis of symmetry is -2.5 and the two points they show here is (-4,-2) and (-1,-2).
a. Use the graph to find the y-intercept. ( The y-intercept is (0,2) )
b. Find the equation of the axis of symmetry. ( x=-5/2 )
c. Use the vertex formula x=-b/2a to find b.
d. Write the equation of the parabola.
e. Test one point using the equation from part (d).
f. Would this method work if the value of a were not known? Explain.
Thanks for taking time from watching the Superbowl to help me out people! Any help is greatly appreciated!
The value of a is 1 because x^2 has a coefficient of 1.
For part c, substitute -5/2 for x (axis of symmetry) to solve for b.
Therefore: -5/2 = -b/2(1); b = 5
For part d, you can determine the equation of the parabola by using one of the points, since you already know b.
y = x^2 + 5x + ?
Using one of the points (-4,-2):
-2 = (-4)^2 + 5(-4) + ?
-2 = 16 - 20 + ?
-2 = -4 + ?
2 = ?
Therefore: y = x^2 + 5x + 2
You can use the point (-1,-2) and achieve the same results.
This would answer part e as well.
This is just one method; there may be other ways to approach the same problem.
I hope this will help.
To solve part (c) and find the value of b, you can use the vertex formula x = -b/2a. The vertex formula gives you the x-coordinate of the vertex of the parabola, which in this case is the axis of symmetry. Since you know the axis of symmetry is -2.5, you can substitute that value in the formula and solve for b.
So the equation becomes:
-2.5 = -b / (2 * 1)
-2.5 = -b/2
To solve for b, you can cross-multiply and solve for b:
-2.5 * 2 = -b
-5 = -b
b = 5
So b is equal to 5.
For part (d), to write the equation of the parabola, you need to use the standard form of the quadratic equation: y = ax^2 + bx + c. Since you know a is 1 (from the given form of the parabola), and you found b as 5, you need to find the value of c.
To find c, you can substitute the coordinates of any of the given points (-4, -2) or (-1, -2) into the equation and solve for c.
Using (-4, -2):
-2 = (-4)^2 + 5(-4) + c
-2 = 16 - 20 + c
-2 = -4 + c
c = 2
So the equation of the parabola is y = x^2 + 5x + 2.
For part (e), you can test the equation by substituting one of the given points into the equation and checking if it satisfies the equation. Let's use (-1, -2):
-2 = (-1)^2 + 5(-1) + 2
-2 = 1 - 5 + 2
-2 = -2
Since -2 = -2, the point (-1, -2) satisfies the equation, confirming that the equation of the parabola is correct.
For part (f), this method of finding b and the equation of the parabola would still work even if the value of a were not known. However, you would need at least two points on the parabola to determine both a and b. The process would be the same - you would substitute the coordinates of two points into the equation and solve for both a and b using a system of equations.