A parabola is defined by the equation 𝑦 = 𝑥2 − 10𝑥 + 16. Determine the coordinates of the vertex
of the parabola.
To find the vertex of the parabola given in this equation, we can first rearrange the equation into vertex form:
𝑦 = 𝑥^2 − 10𝑥 + 16
= (𝑥^2 − 10𝑥 + 25) − 9
= (𝑥 − 5)^2 − 9
The vertex form of a parabola is (h, k), where h is the x-coordinate of the vertex and k is the y-coordinate of the vertex.
In this equation, we can see that the x-coordinate of the vertex is 5 (since the 𝑥 − 5 term is squared and thus always positive). To find the y-coordinate of the vertex, we can simply evaluate the equation when 𝑥 = 5:
𝑦 = (5 − 5)^2 − 9 = -9
Therefore, the vertex of the parabola is at (5, -9).
To find the coordinates of the vertex of a parabola defined by the equation 𝑦 = 𝑥^2 − 10𝑥 + 16, follow these steps:
Step 1: Rewrite the equation in vertex form.
The vertex form of a parabola is given by 𝑦 = 𝑎(𝑥 − ℎ)^2 + 𝑘, where (ℎ, 𝑘) represents the coordinates of the vertex.
In this case, we can rewrite the given equation as 𝑦 = 𝑥^2 − 10𝑥 + 16 by expanding it:
𝑦 = (𝑥^2 − 10𝑥) + 16
Step 2: Complete the square.
To complete the square, we need to add and subtract a constant term to the equation.
The coefficient of 𝑥 in the given equation is -10.
To find the constant term, we divide the coefficient of 𝑥 by 2 and square it: (-10/2)^2 = 25
𝑦 = (𝑥^2 − 10𝑥 + 25 - 25) + 16
𝑦 = (𝑥^2 − 10𝑥 + 25) - 25 + 16
Step 3: Simplify the equation.
𝑦 = (𝑥 - 5)^2 - 9
From the simplified equation, we can see that the vertex is located at (5, -9).
Therefore, the coordinates of the vertex of the given parabola are (5, -9).