Question

what is the equation of the following graph in vertex form?

courtesy of texas instruments

\\( y = (x - 1)^2 \\)

\\( y = (x - 1)^2 + 1 \\)

\\( y = (x + 1)^2 - 1 \\)

\\( y = (x + 1)^2 \\)

Explanation:

Identify the vertex from the graph

The vertex of a parabola is its lowest or highest point. Looking at the given graph, the lowest point (the vertex) is clearly labeled as:

$$(h, k) = (-1, 0)$$

Write the general vertex form equation

The vertex form of a quadratic equation is:

$$y = a(x - h)^2 + k$$

Substituting the vertex coordinates \(h = -1\) and \(k = 0\) into this formula gives:

$$y = a(x - (-1))^2 + 0$$

$$y = a(x + 1)^2$$

Determine the value of the coefficient \(a\)

To find the value of \(a\), we use another point on the parabola. The graph labels the y-intercept at:

$$(x, y) = (0, 1)$$

Substitute these coordinates into our equation:

$$1 = a(0 + 1)^2$$

$$1 = a(1)^2$$

$$a = 1$$

Write the final equation

Substituting \(a = 1\) back into the equation yields:

$$y = (x + 1)^2$$

This matches the fourth option.

Answer:

• (A) \(y = (x - 1)^2\)
• (B) \(y = (x - 1)^2 + 1\)
• (C) \(y = (x + 1)^2 - 1\)
• (D) \(y = (x + 1)^2\) (Correct answer)