Question

What is the y-intercept, axis of symmetry and vertex of the following function.

Answer 1

To find the y-intercept you must substitute 0 into x.

F(x) = -(3x0) - (6x0) - 5
F(x) = -5

Therefore your y intercept is -5.

To find the axis of symmetry and the vertex we can use -b/2a to find the x value.

6/(-3x2) = 6/-6 = -1

Now we know the x value is -1 we can sub it into the equation to find the y value.

F(x) = -(3x-1) - (6x-1) - 5
F(x) = -(-3)- (-6) - 5
F(x) = 3 + 6 - 5
F(x) = 4

Therefore the y value is 4.

The axis of symmetry and vertex is (-1,4) and the y intercept is -5. Hope this helps.

Answer 2

Complete the square:

F(x) = -3x² - 6x - 5

F(x) = -3 (x² + 2x) - 5

F(x) = -3 (x² + 2x + 1 - 1) - 5

F(x) = -3 ((x + 1)² - 1) - 5

F(x) = -3 (x + 1)² + 3 - 5

F(x) = -3 (x + 1)² - 2

The y-intercept has x-coordinate equal to 0, so it corresponds to the value of F(0) :

F(0) = -3 (0 + 1)² - 2 = -3 - 2 = -5

The axis of symmetry is the vertical line running through the vertex of this parabola, so we'll come back to this.

The vertex of the parabola is (-1, -2). This represents the maximum value of F(x), which follows from

(x + 1)² ≥ 0   ⇒   -3 (x + 1)² ≤ 0   ⇒   -3 (x + 1)² - 2 ≤ -2

This is to say, every point on the parabola has a y-coordinate no greater than -2.

As mentioned earlier, the axis of symmetry is the vertical line through the vertex, and its equation is determined by the x-coordinate of the vertex. Hence the AoS is the line x = -1.