Question

the graph above is a transformation of the function \\(f(x) = x^2\\)

write an equation for the function graphed above.

\\(g(x) =\\)

Explanation:

Identify the vertex of the parabola

The vertex is the highest point on this downward-opening parabola.
Looking at the graph, the peak is located at:

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

Write the vertex form equation

The vertex form of a quadratic function is:

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

Substituting the vertex \( (1, -1) \):

$$ g(x) = a(x - 1)^2 - 1 $$

Find the vertical stretch factor

Identify another clear point on the grid that the parabola passes through.
The graph passes through the \(y\)-intercept at:

$$ (0, -2) $$

Substitute \(x = 0\) and \(g(x) = -2\) into the equation:

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

Solve for the coefficient

Simplify the equation to find the value of \(a\):

$$ -2 = a(1) - 1 $$

$$ a = -1 $$

Write the final function

Substitute \(a = -1\) back into the vertex form equation:

$$ g(x) = -(x - 1)^2 - 1 $$

Answer:

The graph above is a transformation of the function \(f(x) = x^2\)

Write an equation for the function graphed above.

\(g(x) =\) <blank>\(-(x - 1)^2 - 1\)</blank>