Find, in the form y = ax^{2} + bx + c, the equation of the quadratic whose graph:
Problem 1 :
cuts the x-axis at 5 and 1, and passes through (2, -9).
Solution:
y = k(x - 5)(x - 1) where k ≠ 0
It passes through (2, -9). Substitute (x, y) = (2, -9)
-9 = k(2 - 5)(2 - 1)
-9 = k(-3)(1)
-9 = -3k
k = 3
By applying k = 3 in above equation
y = 3(x - 5)(x - 1)
y = 3(x^{2} - x - 5x + 5)
y = 3(x^{2} - 6x + 5)
y = 3x^{2} - 18x + 15
Problem 2 :
cuts the x-axis at 2 and -1/2, and passes through (3, -14)
Solution:
It passes through (3, -14). Substitute (x, y) = (3, -14)
By applying k = -4 in above equation
Problem 3 :
touches the x-axis at 3 and passes through (-2, -25)
Solution:
Since the x- intercept is 3(repeated), the equation is
y = k(x - 3)^{2} where k ≠ 0
It passes through (-2, -25). Substitute (x, y) = (-2, -25)
-25 = k(-2 - 3)^{2}
-25 = k(-5)^{2}
25k = -25
k = -1
By applying k = -1 in above equation
y = -1(x - 3)²
y = -(x² - 6x + 9)
y = -x² + 6x - 9
Problem 4 :
touches the x-axis at -2 and passes through (-1, 4)
Solution:
Since the x- intercept is -2(repeated), the equation is
y = k(x + 2)^{2 }
It passes through (-1, 4). Substitute (x, y) = (-1, 4)
4 = k(-1 + 2)^{2}
4 = k(1)^{2}
k = 4
By applying k = 4 in above equation
y = 4(x + 2)²
y = 4(x^{2} + 4x + 4)
y = 4x^{2} + 16x + 16
Problem 5 :
cuts the x-axis at 3, passes through (5, 12) and has axis of symmetry x = 2
Solution:
Since, the x- intercept is 3 and the axis of symmetry is x = 2, the other x- intercept is 1, the equation is
y = k(x - 1)(x - 3) where k ≠ 0
It passes through (5, 12). Substitute (x, y) = (5, 12)
12 = k(5 - 1)(5 - 3)
12 = k(4)(2)
12 = 8k
k = 3/2
By applying k = 3/2 in above equation
Problem 6 :
cuts the x-axis at 5, passes through (2, 5) and has axis of symmetry x = 1
Solution:
Since, the x- intercept is 5 and the axis of symmetry is x = 1, the other x- intercept is -3, the equation is
y = k(x - 5)(x + 3) where k ≠ 0
It passes through (2, 5). Substitute (x, y) = (2, 5)
5 = k(2 - 5)(2 + 3)
5 = k(-3)(5)
5 = -15k
k = -1/3
By applying k = -1/3 in above equation
Mar 14, 24 10:44 PM
Mar 14, 24 10:12 AM
Mar 14, 24 09:52 AM