Question

23: the graph of the function is a translation of the graph of (f(x) = x^2).
graph the function (g(x) = x^2 - 3)

Explanation:

Identify the parent function and transformation

The parent function is given as \(f(x) = x^2\), which represents a standard parabola with its vertex at the origin \((0,0)\). The function to graph is \(g(x) = x^2 - 3\). This is in the form \(g(x) = f(x) + k\), where \(k = -3\).

Determine the vertical shift

A constant subtracted from the output of a function shifts its graph vertically downward. Since \(k = -3\), every point \((x, y)\) on the parent graph \(f(x) = x^2\) is translated down by \(3\) units to become \((x, y - 3)\) on the graph of \(g(x)\).

Calculate key points for the new graph

We can find key points for \(g(x) = x^2 - 3\) by substituting simple integer values for \(x\):

• For \(x = 0\): \(g(0) = 0^2 - 3 = -3\), giving the vertex at \((0, -3)\).
• For \(x = \pm 1\): \(g(\pm 1) = (\pm 1)^2 - 3 = 1 - 3 = -2\), giving points \((1, -2)\) and \((-1, -2)\).
• For \(x = \pm 2\): \(g(\pm 2) = (\pm 2)^2 - 3 = 4 - 3 = 1\), giving points \((2, 1)\) and \((-2, 1)\).
• For \(x = \pm 3\): \(g(\pm 3) = (\pm 3)^2 - 3 = 9 - 3 = 6\), giving points \((3, 6)\) and \((-3, 6)\).

Plot the translated parabola

Using these calculated coordinates, we plot the vertex at \((0, -3)\) and draw a symmetric, upward-opening parabola passing through \((-1, -2)\), \((1, -2)\), \((-2, 1)\), \((2, 1)\), \((-3, 6)\), and \((3, 6)\).

Answer:

To graph the function \(g(x) = x^2 - 3\), translate the parent parabola \(f(x) = x^2\) vertically downward by \(3\) units.

The key points to plot on the coordinate grid are:

• Vertex: \((0, -3)\)
• Points: \((1, -2)\), \((-1, -2)\), \((2, 1)\), \((-2, 1)\), \((3, 6)\), and \((-3, 6)\)