Answer:
x = 9(y - 1)² + 1
see attached for graph
Explanation:
Completing the square
Given equation:
9y² - x - 18y + 10 = 0
As there is a y² in the equation, this implies it is a sideways parabola.
To complete the square for a sideways parabola, collect the y variable terms on the left and everything else on the right:
⇒ 9y² - 18y = x - 10
Factor out the leading coefficient from the left side:
⇒ 9(y² - 2y) = x - 10
Take the coefficient of y, half it, then square it: (-2 ÷ 2)² = 1
Add this inside the parentheses, and add the distributed value of this to the right side:
⇒ 9(y² - 2y + 1) = x - 10 + 9
Factor the parentheses and simplify the right side:
⇒ 9(y - 1)² = x - 1
Add 1 to both sides to make x the subject:
⇒ x = 9(y - 1)² + 1
Graphing the parabola
Conic form of a sideways parabola with a horizontal axis of symmetry:
x = a(y - k)² + h
(where (h, k) is the vertex and y = k is the axis of symmetry)
If a > 0, the parabola opens to the right, and if a < 0, the parabola opens to the left.
Therefore, for x = 9(y - 1)² + 1
- vertex = (1, 1)
- axis of symmetry: y = 1
- a > 0 ⇒ the parabola opens to the right
To find the x-intercept, set y = 0 and solve for x:
⇒ x = 9(0 - 1)² + 1
⇒ x = 9(-1)² + 1
⇒ x = 9(1) + 1
⇒ x = 10
Therefore, the y-intercept is (10, 0)
As the axis of symmetry is y = 1, the other value of y when x = 10 is y = 2
Plot the vertex and points. Add an axis of symmetry to help draw the curve. Draw the parabola opening to the right.
**graph attached**