How do you recognize the graph of y = (x-2)(x+2)?

You recognise the graph of \( y = (x-2)(x+2) \) as a parabola opening upwards with roots at \( x = -2 \) and \( x = 2 \).

To understand this better, let's break down the equation \( y = (x-2)(x+2) \). This is a quadratic equation in its factorised form. When you expand it, you get \( y = x^2 - 4 \). This tells us that the graph is a parabola, a U-shaped curve, because the highest power of \( x \) is 2.

The roots of the equation, also known as the x-intercepts, are the values of \( x \) that make \( y = 0 \). Setting \( y = 0 \) in the equation \( (x-2)(x+2) = 0 \), we find that \( x = 2 \) and \( x = -2 \). These are the points where the graph crosses the x-axis.

The vertex of the parabola, which is its lowest point since it opens upwards, is exactly halfway between the roots. The midpoint between \( x = -2 \) and \( x = 2 \) is \( x = 0 \). Substituting \( x = 0 \) into the equation \( y = x^2 - 4 \), we get \( y = -4 \). So, the vertex is at \( (0, -4) \).

The graph is symmetric about the y-axis because the equation \( y = x^2 - 4 \) is unchanged if \( x \) is replaced by \(-x\). This symmetry helps in sketching the graph accurately. The y-intercept, where the graph crosses the y-axis, is found by setting \( x = 0 \), giving \( y = -4 \).

In summary, the graph of \( y = (x-2)(x+2) \) is a parabola opening upwards, with roots at \( x = -2 \) and \( x = 2 \), a vertex at \( (0, -4) \), and symmetry about the y-axis.