Question

The vertex of this parabola is at (-2,-3). When the y-value is -2, the x-value is -5. What is the coefficient of the squared term in the parabola's equation? 5 re (-2, -3)​

Answer 1

The coefficient of the squared term of the equation is 1/9.

Explanation:

We are given that the vertex of the parabola is at (-2, -3). We also know that when the y-value is -2, the x-value is -5. Using this information we want to find the cofficient of the squared term in the parabola's equation.

Since we are given the vertex, we can use the vertex form:

`\displaystyle y=a(x-h)^2+k`

Where a is the leading coefficient and (h, k) is the vertex.

Since the vertex is (-2, -3), h = -2 and k = -3:

`\displaystyle y=a(x-(-2))^2+(-3)`

Simplify:

`y=a(x+2)^2-3`

We are also given that y = -2 when x = -5. Substitute:

`(-2)=a(-5+2)^2-3`

Solve for a. Simplify:

`\displaystyle \begin{aligned} -2&=a(-3)^2-3\\ 1&=9a \\a&=(1)/(9)\end{aligned}`

Therefore, our full vertex equation is:

`\displaystyle y=(1)/(9)(x+2)^2-3`

We can expand:

`\displaystyle y=(1)/(9)(x^2+4x+4)-3`

Simplify:

`\displaystyle y=(1)/(9)x^2+(4)/(9)x-(23)/(9)`

The coefficient of the squared term of the equation is 1/9.