Consider the function f(x)=-4x^2-8x+10. Determine whether the graph opens up or down. Find the axis of symmetry, the vertex, and the y-intercept. Graph the function.

Answer 1

Since the coefficient of the x² term is negative, the graph open downward.

The axis of symmetry is at -b/2a, i.e. x=-1

The y intercept (i.e. where the graph crosses the y axis) is where x = 0 and y=10.

The vertex is (-1,14)

Answer 2

First, look at the dominant function: y = -x² . For large x ( + or - ), y is a large negative value. Therefore the curve (a Parabola) opens down.

To find the location of various features, we can put the equation into an "offset" format:

K(y + B )= (x + A)²

Divide the given equation by -4 to get:

x² + 2x - 2.5 = -y/4

move the constant to the other side and then "complete the square" on the x side:

x² + 2x = -y/4 + 2.5

x² + 2x + 1 = -y/4 + 2.5 + 1

(x + 1)² = -y/4 + 3.5

Finally, put into the general format:

(x + 1)² = -1/4 * (y - 14)

Now, we can see that the axis of symmetry is x = -1. To find the vertex, find y at x = -1:

-1/4 * (y - 14) = 0

y/4 = 3.5

y = 14

CHECK by plotting the curve ( I did it using the desmos online tool)