The equation of the parabola could be written as y-k = a(x-h)^2, where (h,k) is the vertex. Thus, y-(-3) = a(x+4)^2, or y+3 = a(x+4)^2.
The coordinates of one x-intercept are (-11,0). Thus, y+3 = a(x+4)^2 becomes 0+3 = a(-11+4)^2, so that 3 = a(-7)^2, or 3 = 49a. Therefore, a = 3/49, and the equation of the parabola becomes
y+3 = (3/49)(x+4)^2.
To find the other x-intercept, let y = 0 and solve the resulting equation for x:
0+3 = (3/49)(x+4)^2, or (49/3)*2 = (x+4)^2
Taking the sqrt of both sides, plus or minus 49/3 = x+4.
plus 49/3 = x+4 results in 37/3 = x, whereas
minus 49/3 = x+4 results in x = -61/3. Unfortunatelyi, this disagrees with what we are told: that one x-intercept is x= -11, or (-11,0).
Trying again, using the quadratic equation y=ax^2 + bx + c, we substitute the coordinates of the points (-4,-3) and (-11,0) and solve for {a, b, c}:
-3 = a(-4)^2 + b(-4) + c, or -3 = 16a - 4b + c
0 = a(-11)^2 - 11b + c, or 0 = 121a - 11b + c
If the vertex is at (-4,-3), then, because x= -b/(2a) also represents the x-coordinate of the vertex, -4 = b / (2a), or -8a = b, or 0 = 8a + b
Now we have 3 equations in 3 unknowns:
0 = 8a + 1b -3 = 16a - 4b + c 0 = 121a - 11b + c
This system of 3 linear equations can be solved in various ways. I've used matrices, finding that a, b and c are all zero. This is wrong.
So, let's try again. Recall that x = -b / (2a) is the axis of symmetry, which in this case is x = -4. If one zero is at -11, this point is 7 units to the left of x = -4. The other zero is 7 units to the right of x = -4, that is, at x = 3.
Now we have 3 points on the parabola: (-11,0), (-4,-3) and (3,0).
This is sufficient info for us to determine {a,b,c} in y=ax^2+bx+c. One by one we take these 3 points and subst. their coordinates into y=ax^2+bx+c, obtaining 3 linear equations:
Each student club must contribute $ 33.33 in order to meet the fundraising goal.
Step-by-step explanation:
Given that a school fundraiser has a minimum target of $ 500. Faculty have donated $ 100 and there are 12 student clubs that are participating with different activities, to determine how much money should each club raise to meet the fundraising goal, the following calculation must be performed:
(500 - 100) / 12 = X
400/12 = X
33,333 = X
Thus, each student club must contribute $ 33.33 in order to meet the fundraising goal.
One way to tackle this is by getting another sheet of paper and drawing it out, then counting up the total of the sides. If you draw it, you can see that you're dealing with a rectangle; two sides of length 12 and two sides of length 8. If you don't like drawing or don't want to in this case, another way to get the answer is by knowing one vertex is at (0, 0), so the next vertex (0, 8), would create a side that's exactly 8 units long. Kind of the same, you know from (0, 0), you also have a point (12, 0), so drawing that would create a side that's 12 units long. All in all, to get the perimeter in units, you have 12 + 12 + 8 + 8 = 40.
The problem says it wants the amount of wood in centimeters needed for the perimeter. What we just found was the perimeter in generic units, so if the problem says every "grid square", or unit, is 3 centimeters long, then all you have to do is take our result 40 and multiply it by 3 to get the number of centimeters. Your perimeter in centimeters would be 120 cm.