Question

solve the system of equations by graphing. first graph the equations, and then fill in the solutions.
$y = \frac{1}{2}(x - 2)^2 - 8$
$y = x - 6$
to graph a quadratic equation, click to plot the vertex first.
to graph a linear equation, click to plot points on the graph.
the solutions are (\square,\square) and (\square,\square).

Explanation:

Step1: Set equations equal

Since \( y \) is expressed in both equations, set \( \frac{1}{2}(x - 2)^2 - 8 = x - 6 \).

Step2: Simplify the equation

Multiply both sides by 2: \( (x - 2)^2 - 16 = 2x - 12 \). Expand \( (x - 2)^2 \): \( x^2 - 4x + 4 - 16 = 2x - 12 \). Simplify: \( x^2 - 4x - 12 = 2x - 12 \). Subtract \( 2x - 12 \) from both sides: \( x^2 - 6x = 0 \). Factor: \( x(x - 6) = 0 \).

Step3: Solve for x

Set each factor to zero: \( x = 0 \) or \( x - 6 = 0 \Rightarrow x = 6 \).

Step4: Find corresponding y

For \( x = 0 \), use \( y = x - 6 \): \( y = 0 - 6 = -6 \). For \( x = 6 \), \( y = 6 - 6 = 0 \).

Answer:

The solutions are \((0, -6)\) and \((6, 0)\).