To find the requested values for the given parabola f(x) = x^2 - 10x + 2, let's go through each question:
A) The value of f(-5):
To find f(-5), substitute x = -5 into the equation:
f(-5) = (-5)^2 - 10(-5) + 2
f(-5) = 25 + 50 + 2
f(-5) = 77
Therefore, the value of f(-5) is 77.
B) The vertex:
The vertex of a parabola can be found using the formula x = -b/2a and substituting it into the equation to find the corresponding y-value.
For the given parabola f(x) = x^2 - 10x + 2:
a = 1, b = -10
x = -(-10) / (2 * 1)
x = 10 / 2
x = 5
Substituting x = 5 into the equation:
f(5) = (5)^2 - 10(5) + 2
f(5) = 25 - 50 + 2
f(5) = -23
Therefore, the vertex is (5, -23).
C) The y-intercept:
The y-intercept occurs when x = 0. Substitute x = 0 into the equation to find the corresponding y-value:
f(0) = (0)^2 - 10(0) + 2
f(0) = 0 - 0 + 2
f(0) = 2
Therefore, the y-intercept is the point (0, 2).
D) The values of x that make f(x) = 0:
To find the values of x that make f(x) = 0, we can solve the quadratic equation x^2 - 10x + 2 = 0. We can either factor the equation or use the quadratic formula.
Using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For the equation x^2 - 10x + 2 = 0:
a = 1, b = -10, c = 2
x = (-(-10) ± √((-10)^2 - 4(1)(2))) / (2(1))
x = (10 ± √(100 - 8)) / 2
x = (10 ± √92) / 2
x = (10 ± 2√23) / 2
x = 5 ± √23
Therefore, the two values of x that make f(x) = 0 are approximately x = 5 + √23 and x = 5 - √23.