The parabola y = x²+6x+8 opens upwards, vertex at (-3,1), intersects x-axis at (-4,0) and (-2,0).
The graph of the function y = x² + 6x + 8 is a parabola that opens upwards. The vertex of the parabola is at the point (-3, 1), and the y-intercept is at the point (0, 8). The parabola intersects the x-axis at the points (-4, 0) and (-2, 0).
To analyze the graph of the function, we can use the following steps:
1. Find the vertex of the parabola.** The vertex of a parabola is the point where the parabola changes direction. To find the vertex, we can use the following formula:
x-coordinate of vertex = -b/2a
y-coordinate of vertex = f(-b/2a)
In this case, a = 1 and b = 6, so the x-coordinate of the vertex is -6/2 = -3. The y-coordinate of the vertex is f(-3) = (-3)² + 6(-3) + 8 = 1. Therefore, the vertex of the parabola is at the point (-3, 1).
2. Find the y-intercept of the parabola.** The y-intercept of a parabola is the point where the parabola intersects the y-axis. To find the y-intercept, we can simply set x to zero and evaluate the function. In this case, f(0) = 0² + 6(0) + 8 = 8. Therefore, the y-intercept of the parabola is at the point (0, 8).
3. Find the x-intercepts of the parabola.** The x-intercepts of a parabola are the points where the parabola intersects the x-axis. To find the x-intercepts, we can set y to zero and solve the resulting equation. In this case, the equation is x² + 6x + 8 = 0. We can factor this equation as (x + 4)(x + 2) = 0. Therefore, the x-intercepts of the parabola are at the points (-4, 0) and (-2, 0).
4. Analyze the graph of the function.** Based on the information we have gathered above, we can conclude that the graph of the function y = x² + 6x + 8 is a parabola that opens upwards. The vertex of the parabola is at the point (-3, 1), and the y-intercept is at the point (0, 8). The parabola intersects the x-axis at the points (-4, 0) and (-2, 0).
Here is a more detailed analysis of the graph of the function:
The parabola opens upwards because the coefficient of the x² term is positive.
The vertex of the parabola is at the point (-3, 1) because the parabola changes direction at this point.
The y-intercept of the parabola is at the point (0, 8) because the parabola intersects the y-axis at this point.
The parabola intersects the x-axis at the points (-4, 0) and (-2, 0) because the parabola has roots at these points.