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use the graph to determine the number and type of solutions of the quadratic equation​

Please help: use the graph to determine the number and type of solutions of the quadratic-example-1

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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.

User Spencer Rohan
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