Question

a quadratic function on a graph has the vertex at the point $(-3, -2)$. which of the following transformations translates the vertex to the point $(0, 0)$? (1 point)
shift left 3 units and up 2 units
shift right 3 units and up 2 units
shift right 3 units and down 2 units
shift left 2 units and down 3 units

Explanation:

Step1: Analyze x - coordinate change

The original x - coordinate of the vertex is \(-3\), and we want to get to \(x = 0\). To go from \(-3\) to \(0\), we use the formula for horizontal shifts: if we have a point \((x,y)\) and we want to shift it horizontally by \(h\) units, the new \(x\) - coordinate is \(x+h\). We want \(x + h=0\) when \(x=-3\), so \(-3 + h=0\), which gives \(h = 3\). A positive \(h\) means a shift to the right by 3 units.

Step2: Analyze y - coordinate change

The original y - coordinate of the vertex is \(-2\), and we want to get to \(y = 0\). Using the formula for vertical shifts: if we have a point \((x,y)\) and we want to shift it vertically by \(k\) units, the new \(y\) - coordinate is \(y + k\). We want \(y + k=0\) when \(y=-2\), so \(-2 + k=0\), which gives \(k = 2\). A positive \(k\) means a shift up by 2 units.

Answer:

shift right 3 units and up 2 units