Question

Investigate EACH of the four parabolas below by graphing them all on one set of axes.

Describe everything you can about the parabolas, including any connections you notice between the rules and the graphs. In your description, you can refer to the equation by the letter in the table.

a.

y=x2+6

b.

y=−x2+6

c.

y=x2+5x+6

d.

y=x2−x−6

Relevant vocabulary that your answer may include: x-intercepts, y-intercept, vertex, and "opening up"/"opening down."

Answer 1

Please find the graphs of the parabola created with MS Excel

The descriptions are as follows, using the reference letter for each of the parabola

a. Parabola 'a' y = x² + 6

Does not intercept the x-axis

y-intercept is y = 6

The vertex is (6, 0)

Opens up

b. Parabola 'b' y = -x² + 6

x-intercepts are at ±√3

y-intercept is y = 6

The vertex is at (0, 6)

Opens down

c. Parabola 'c' x² + 5·x + 6

x-intercepts are at x = -2, and -3, (-2, 0), and (-3, 0)

y-intercept is y = 6, (0, 6)

The vertex of parabola 'c' is (-2.5, -0.25)

Opens up

d. Parabola 'd' x² - x - 6

x-intercept at x = -2 (-2, 0)

y-intercept is y = -6

The vertex of parabola is at (-0.5, 6.75)

Opens up

Explanation:

a. Parabola 'a' y = x² + 6 does not have an x-intercept

The y-intercept of parabola a is y = 6

The vertex of parabola 'a' is at (6, 0)

Parabola 'a' opens up

b. Parabola 'b' y = -x² + 6 has an x-intercept at ±√3

The y-intercept of parabola b is y = 6

The vertex of parabola 'b'

At the vertex, x = -0/2 = 0

y = (0)² + 5×(0) + 6 = 6

The vertex of parabola 'b' is at (0, 6)

Parabola 'b' opens down

c. Parabola 'c' x² + 5·x + 6 has an x-intercept at x = -2, and -3

The y-intercept of parabola 'c' is y = 6

The vertex of parabola 'c'

At the vertex, x = -5/2 = -2.5

y = (-2.5)² + 5×(-2.5) + 6 = -0.25

The vertex of parabola 'c' is at (-2.5, -0.25)

Parabola 'c' opens up

d. Parabola 'd' y = x² - x - 6 has an x-intercept at x = -2

The y-intercept of parabola 'd' is y = -6

The vertex of parabola 'd'

At the vertex, x = -1/2 = -0.5

y = (-0.5)² - (-0.5) + 6 = 6.75

The vertex of parabola 'd' is at (-0.5, 6.75)

Parabola 'd' opens up