Identify the vertex, intercepts, and zeros of the function.
g(x) = x^2 + 8x + 11
Please help!!
To identify the vertex, intercepts, and zeros of the function g(x) = x^2 + 8x + 11, you can use the quadratic formula or complete the square to find the vertex and zeros.
To find the vertex of a quadratic function in the form ax^2 + bx + c, you can use the formula x = -b/(2a). In this case, a = 1, b = 8, and c = 11.
1. Vertex: Use the formula x = -b/(2a) = -8/(2*1) = -8/2 = -4. So the x-coordinate of the vertex is -4. To find the y-coordinate, substitute this x-value back into the original equation:
g(-4) = (-4)^2 + 8(-4) + 11 = 16 - 32 + 11 = -5. So the vertex is (-4, -5).
2. x-intercepts: The x-intercepts are the points where the graph of the function crosses the x-axis. To find the x-intercepts, you set g(x) = 0 and solve for x.
Setting g(x) = 0, we have x^2 + 8x + 11 = 0. This equation cannot be factored easily, so we can either use the quadratic formula or complete the square.
Using the quadratic formula, x = (-b ± √(b^2 - 4ac)) / (2a). Plugging in the values of a = 1, b = 8, and c = 11, we get:
x = (-8 ± √(8^2 - 4(1)(11))) / (2*1)
x = (-8 ± √(64 - 44)) / 2
x = (-8 ± √20) / 2
x = (-8 ± 2√5) / 2
x = -4 ± √5.
So the x-intercepts are x = -4 + √5 and x = -4 - √5.
3. y-intercept: The y-intercept is the point where the graph crosses the y-axis. To find the y-intercept, set x = 0 and solve for y:
g(0) = (0)^2 + 8(0) + 11 = 11. So the y-intercept is (0, 11).
Overall:
- Vertex: (-4, -5)
- x-intercepts: -4 + √5 and -4 - √5
- y-intercept: (0, 11)