Question

Graph the parabola y = x^2 + 6x + 9 by plotting the vertex, the x and y-intercepts, and the point symmetric to intercept across the axis of symmetry.

Answer 1

x intercept and turning/minimum point: (-3,0)

y intercept: (0,9)

Attached image contains the graph of the equation.

Explanation:

At the x intercept, y = 0.

At the y intercept, x = 0.

Find the x intercept(s) by replacing y with 0:

x^2 + 6x +9 = 0

Solve for x using the quadratic formula:

x = (-b ± sqrt(b^2 - 4ac)) / 2a

x = (-6 ± sqrt(6^2 - 4 * 1 * 9)) / 2 * 1

This gives the following single answer:

x = -3

As there is only one solution , we know it is a repeated root. This means the turning point of the curve is also here. We also know this turning point is a minimum point as opposed to a maximum point, as the coefficient of x^2 is positive.

The y intercept can be found by substituting x for 0:

y = 0^2 + 6(0) + 9

y = 9

Now we know the x and y intercepts:

(-3,0) and (0,9)

Therefore we can sketch the curve accordingly.