Find the vertex AND factored form equation for both situations
1) A parabola has vertex (-1, 3) and goes through point (2, 5).
2) Write the equation for the parabola with x-intercepts (0, 1) and (0, -3) and stretched by a factor 4.
#1. Given the vertex, the equation is y = a(x+1)^2 + 3
now plug in the point to find a.
#2. Given the intercepts, we have
x = a(y-1)(y+3)
You don't say whether the stretch is horizontal or vertical.
To find the vertex and factored form equation for each situation, we can use the general equation for a parabola, which is given by y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola.
1) For the first situation, we are given the vertex (-1, 3) and a point it goes through (2, 5).
a) To find the value of 'a', we can substitute the coordinates of the point (2, 5) into the equation:
5 = a(2 - (-1))^2 + 3
b) Next, we can calculate the value of 'a':
5 = a(3)^2 + 3
5 = 9a + 3
9a = 5 - 3
9a = 2
a = 2/9
c) Now that we have the value of 'a', we can substitute it into the equation and simplify to find the factored form:
y = (2/9)(x - (-1))^2 + 3
y = (2/9)(x + 1)^2 + 3
y = (2/9)(x^2 + 2x + 1) + 3
y = (2/9)x^2 + (4/9)x + 2/9 + 3
y = (2/9)x^2 + (4/9)x + (20/9)
Therefore, the factored form equation for the parabola is y = (2/9)x^2 + (4/9)x + (20/9), and the vertex is (-1, 3).
2) For the second situation, we are given the x-intercepts (0, 1) and (0, -3), and the parabola is stretched by a factor of 4.
a) To find the value of 'a', we can use the x-intercepts in the equation:
y = a(x - x1)(x - x2)
Substituting the values of the x-intercepts (0, 1) and (0, -3) into the equation:
1 = a(0 - 0)(0 - (-3))
1 = a(0)(3)
1 = 0
This means that 'a' is equal to 0, which implies that the parabola is actually a straight line.
b) Since the parabola is actually a line, we can write the equation in slope-intercept form:
y = mx + b
The slope 'm' can be found using the given x-intercepts:
m = (y2 - y1)/(x2 - x1)
m = (-3 - 1)/(0 - 0)
m = -4/0
As m is undefined, this implies that the slope of the line is vertical.
c) Finally, using the point-slope form of a vertical line, we can determine the equation:
x = 0
Therefore, the equation for the parabola with x-intercepts (0, 1) and (0, -3) stretched by a factor of 4 is x = 0, which represents a vertical line passing through the y-axis.