To sketch the graph of the quadratic function y = -x^2 - x + 6 , we need to find the intercepts, the vertex, and determine the concavity of the parabola.

Finding the y-intercept

The y-intercept occurs where the graph crosses the y-axis, which is when x = 0 .

Substitute x = 0 into the equation:

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

Thus, the y-intercept is at the point (0, 6).

Finding the x-intercepts

The x-intercepts occur where the graph crosses the x-axis, which is when y = 0 .

Set the equation to zero and solve for x :

0 = -x^2 - x + 6

This is a quadratic equation in the form ax^2 + bx + c = 0 , where a = -1 , b = -1 , and c = 6 .

Use the quadratic formula to find the roots:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Substitute the values of a , b , and c :

x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(-1)(6)}}{2(-1)}

x = \frac{1 \pm \sqrt{1 + 24}}{-2}

x = \frac{1 \pm \sqrt{25}}{-2}

x = \frac{1 \pm 5}{-2}

This gives two solutions:

x = \frac{1 + 5}{-2} = -3 \quad \text{and} \quad x = \frac{1 - 5}{-2} = 2

Thus, the x-intercepts are at the points (-3, 0) and (2, 0).

Finding the vertex

The vertex of a parabola in the form y = ax^2 + bx + c can be found using the formula for the x-coordinate of the vertex:

x = -\frac{b}{2a} \,\,\,\,\,\text{SL 2.6}

Substitute a = -1 and b = -1 :

x=-\frac{-1}{2(-1)}=-\frac12

Substitute x=-\frac12 back into the original equation to find the y-coordinate of the vertex:

y=-\left(\frac12\right)^2+\frac12+6

y=-\frac14+\frac12+6

y=-\frac14+\frac24+6

y=\frac14+6

y=\frac{1+24}{4}

y=\frac{25}{4}

Thus, the vertex is at the point \left(-\frac12,\frac{25}{4}\right).

Determining the concavity

The concavity of a parabola is determined by the sign of the coefficient a in the quadratic equation y = ax^2 + bx + c .

If a > 0 , the parabola is concave up.
If a < 0 , the parabola is concave down.

In this case, a = -1 , which is less than zero. Therefore, the parabola is concave down.

Summary

y-intercept: (0, 6)
x-intercepts: (-3, 0) and (2, 0)
Vertex: \left(\frac12,\frac{25}{4}\right)
Concavity: The parabola is concave down.

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