Answers:
Standard form: f(x) = (x - 3)² - 16
vertex: (3, -16)
x-intercept (x, y) = (-1, 0)
x-intercept (x, y) = (7, 0)
y-intercept (x, y) = (0, -7)
Step-by-step explanation:
The given equation is
f(x) = x² - 6x - 7
To write in the standard form, we need to add and subtract (b/2)², where b is the number besides x, so replacing b = -6, we get:
(b/2)² = (-6/2)² = (-3)² = 9
Then, when we add and subtract 9, we get:
f(x) = x² - 6x + 9 - 9 - 7
f(x) = (x - 3)² - 16
Now, in an equation of the form y = (x - h)² + k, the vertex is (h, k). So, in this case, the vertex is (3, -16)
On the other hand, the y-intercept is the point where the line crosses the y-axis. It can be calculated replacing x by 0 on the equation of f(x), so
f(x) = x² - 6x - 7
f(0) = 0² - 6(0) - 7
f(0) = -7
Therefore, the y-intercept is the point (0, -7)
To find the x-intercepts, we need to make f(x) = 0, so
f(x) = x² - 6x - 7 = 0
(x - 7)(x + 1) = 0
Then
x - 7 = 0
x - 7 + 7 = 0 + 7
x = 7
and
x + 1 = 0
x + 1 - 1 = 0 - 1
x = -1
Therefore, the x-intercepts are (7, 0) and (-1, 0).
So, the answers are
Standard form: f(x) = (x - 3)² - 16
vertex: (3, -16)
x-intercept (x, y) = (-1, 0)
x-intercept (x, y) = (7, 0)
y-intercept (x, y) = (0, -7)
Then, the graph of the function is