17-Identify the vertex, axis of symmetry and x-intercept(s) of F(x)=(x+5)^2-6. I have no idea how the x intercepts are -5 +/- sq. root 6 and 0. can you show me how this came to be? I factored it so it would be x^2+10x+19, but there's no common factors in ten and 19. Can you show me, please?
for the x-intercepts, F(x) = 0. (the y = 0)
so (x+5)^2-6 = 0
(x+5)^2 = 6
x+5 = ± √6
x = -5 ± √6
You expanded the function, you did not factor it.
Once you get x^2 + 10x + 19 = 0,
if you use the formula and simplify the answer, it comes out the same as my answer.
I have no idea where you got the 0, a parabola can have at most 2 x-intercepts.
Sure! Let's break down the process step by step.
To find the vertex, axis of symmetry, and x-intercepts of the quadratic function F(x) = (x + 5)^2 - 6, we need to rewrite the equation in standard form (ax^2 + bx + c) and apply some quadratic formula.
1. Standard Form:
F(x) = (x + 5)^2 - 6
=> F(x) = (x^2 + 10x + 25) - 6
=> F(x) = x^2 + 10x + 19
Now, let's find the vertex, axis of symmetry, and x-intercepts:
Vertex:
1. The vertex form of a quadratic equation is given by: f(x) = a(x - h)^2 + k, where (h, k) represents the vertex.
In our equation, F(x) = x^2 + 10x + 19, which is already in standard form. So, we don't need to convert it.
2. The vertex is given by (h, k), where h = -b/2a.
In our equation, a = 1, and b = 10.
So, the vertex x-coordinate (h) = -10/(2*1) = -5.
3. Now, we substitute the x-coordinate h into the equation to find the y-coordinate (k).
F(x) = (-5)^2 + 10(-5) + 19
=> F(x) = 25 - 50 + 19
=> F(x) = -6.
Thus, the vertex is (-5, -6).
Axis of Symmetry:
The axis of symmetry is the vertical line that passes through the vertex. In this case, the axis of symmetry is x = -5.
X-intercepts:
To find the x-intercepts, we set F(x) = 0 and solve for x.
1. Set F(x) = x^2 + 10x + 19 = 0.
2. Now, we apply the quadratic formula for finding the solutions (x-intercepts):
x = (-b ± √(b^2 - 4ac)) / 2a.
In our equation, a = 1, b = 10, and c = 19.
3. Substitute these values into the quadratic formula:
x = (-10 ± √(10^2 - 4 * 1 * 19)) / (2 * 1)
x = (-10 ± √(100 - 76)) / 2
x = (-10 ± √24) / 2
4. Simplify further:
x = (-10 ± √(4 * 6)) / 2
x = (-10 ± 2√6) / 2
x = -5 ± √6
The x-intercepts are -5 + √6 and -5 - √6.
Therefore, the vertex is (-5, -6), the axis of symmetry is x = -5, and the x-intercepts are -5 + √6 and -5 - √6.