Question

vertex (3, 6), y-intercept 2.

Answer 1

Given vertex = (3,6)

y - intercept point (0,2)

Line equation through the two points is

Cross multiplication.

3y - 18 = 4x - 12

Add 18 to each side.

3y  = 4x - 12 + 18

3y  = 4x + 6

Divide each side by 3.

Required line equation y  = (4/3)x + 2.

Comments

Slope-intercept form line equation is y = mx + b, where m is slope and b is y-intercept.

Answer 2

Slope-intercept form line equation is y = mx + b, where m is slope and b is y-intercept.

The vertex is (3, 6) and the y - intercept is 2.

y - intercept is 2.

⇒ b = 2.

Now, the line equation is y = mx + 2.

Find the slope by substituting the point in the line equation say (x, y) = (3, 6).

6 = m(3) + 2

3m = 6 - 2

3m = 4

⇒ m = 4/3.

The line equation in slope - intercept - form is y = (4/3)x + 2.

Answer 3

Vertex is (3, 6) and y - intercept is 2.

In this question, there is no mention to what find out by using given data, so here find any solution which is related to the given data.

Here vertex is mentioned, so find the any solution for the conic section.

Here the vertex is only one and the only parabola contains single vertex in the conic section.

So, find the equation of parabola by using given data.

The standard form of the equation of a parabola with vertex at (h, k) and directrix y = k - p is (x - h)² = 4p(y - k), p ≠ 0 (Vertical axis). For directrix x = h - p the equation is (y - k)² = 4p(x - h), p ≠ 0 (Horizontal axis). The focus is on the axis p units (directed distance) from the vertex.

Vertex = (h, k) = (3, 6) and y - intercept is 2, so the parabola crosses y - axis at (0, 2).

Case 1 : Vertical axis : (x - h)² = 4p(y - k).

The equation of parabola is (x - 3)² = 4p(y - 6).

To find the p value, substitute the values of (x, y) = (0, 2) in the above equation.

[ (0) - 3 ]² = 4p[ (2) - 6 ]

9 = 4p(- 4)

9 = - 16p

p = - 9/16.

The vertical axis parabola equation is (x - 3)² = 4(- 9/16)(y - 6).

Case 2 : Horizontal axis : (y - k)² = 4p(x - h).

The equation of parabola is (y - 6)² = 4p(x - 3).

To find the p value, substitute the values of (x, y) = (0, 2) in the above equation.

[ (2) - 6 ]² = 4p[ (0) - 3 ]

(- 4)² = 4p(- 3)

16 = - 12p

p = - 16/12 = - 4/3.

The vertical axis parabola equation is (y - 6)² = 4(- 4/3)(x - 3).

In this case, there are two y - intercepts are available since y - term is squared.

The parabola equations are (x - 3)² = - 9/4 (y - 6) or (y - 6)² = - 16/3 (x - 3).