Find the equation of the quadratic with graph:
Problem 1 :
Solution:
Intercept form equation of the above parabola:
y = a(x - p)(x - q)
Because x-intercepts are (1, 0) and (2, 0).
x = 1 ---> x - 1 = 0
x = 2 ---> x - 2 = 0
Then,
y = a(x - 1)(x - 2)
It passes through (0, 4). Substitute (x, y) = (0, 4).
4 = a(0 - 1)(0 - 2)
4 = a(-1)(-2)
4 = 2a
a = 2
Intercept form equation of the parabola:
y = 2(x - 1)(x - 2)
Problem 2 :
Solution:
Vertex form equation of the above parabola:
y = a(x - h)^{2} + k
Vertex (h, k) = (2, 0)
y = a(x - 2)^{2} + 0
y = a(x - 2)^{2}
It passes through (0, 8). Substitute (x, y) = (0, 8).
8 = a(0 - 2)^{2}
8 = a(-2)^{2}
8 = 4a
a = 2
Vertex form equation of the parabola:
y = 2(x - 2)^{2}
Problem 3 :
Solution:
Intercept form equation of the above parabola:
y = a(x - p)(x - q)
Because x-intercepts are (1, 0) and (3, 0).
x = 1 ---> x - 1 = 0
x = 3 ---> x - 3 = 0
Then,
y = a(x - 1)(x - 3)
It passes through (0, 3). Substitute (x, y) = (0, 3).
3 = a(0 - 1)(0 - 3)
3 = a(-1)(-3)
3 = 3a
a = 1
Intercept form equation of the parabola:
y = 1(x - 1)(x - 3)
Problem 4 :
Solution:
Intercept form equation of the above parabola:
y = a(x - p)(x - q)
Because x-intercepts are (-1, 0) and (3, 0).
x = -1 ---> x + 1 = 0
x = 3 ---> x - 3 = 0
Then,
y = a(x + 1)(x - 3)
It passes through (0, 3). Substitute (x, y) = (0, 3).
3 = a(0 + 1)(0 - 3)
3 = a(1)(-3)
3 = -3a
a = -1
Intercept form equation of the parabola:
y = -1(x + 1)(x - 3)
Problem 5 :
Solution:
Vertex form equation of the above parabola:
y = a(x - h)^{2} + k
Vertex (h, k) = (1, 0)
y = a(x - 1)^{2} + 0
y = a(x - 1)^{2}
It passes through (0, -3). Substitute (x, y) = (0, -3).
-3 = a(0 - 1)^{2}
-3 = a(-1)^{2}
-3 = a
a = -3
Vertex form equation of the parabola:
y = -3(x - 1)^{2}
Problem 6 :
Solution:
Intercept form equation of the above parabola:
y = a(x - p)(x - q)
Because x-intercepts are (-2, 0) and (3, 0).
x = -2 ---> x + 2 = 0
x = 3 ---> x - 3 = 0
Then,
y = a(x + 2)(x - 3)
It passes through (0, 12). Substitute (x, y) = (0, 12).
12 = a(0 + 2)(0 - 3)
12 = a(2)(-3)
12 = -6a
a = -2
Intercept form equation of the parabola:
y = -2(x + 2)(x - 3)
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