Question

21/41 graph y = (x - 3)^2 - 4

Explanation:

Step1: Identify the vertex form

The equation \( y=(x - 3)^2-4 \) is in vertex form of a parabola, which is \( y = a(x - h)^2 + k \), where \((h,k)\) is the vertex. Here, \( h = 3 \) and \( k=-4 \), so the vertex is \((3,-4)\).

Step2: Determine the direction of opening

Since \( a = 1>0 \), the parabola opens upwards.

Step3: Find the axis of symmetry

The axis of symmetry for a parabola in vertex form \( y = a(x - h)^2 + k \) is the vertical line \( x = h \). So, the axis of symmetry is \( x = 3 \).

Step4: Find the y - intercept

To find the y - intercept, set \( x = 0 \) in the equation:
\( y=(0 - 3)^2-4=9 - 4 = 5 \). So, the y - intercept is \((0,5)\).

Step5: Find the x - intercepts

To find the x - intercepts, set \( y = 0 \):
\( 0=(x - 3)^2-4 \)
\((x - 3)^2=4\)
Take square roots on both sides: \( x - 3=\pm2 \)
Case 1: \( x - 3 = 2\Rightarrow x=2 + 3=5 \)
Case 2: \( x - 3=-2\Rightarrow x=-2 + 3 = 1 \)
So, the x - intercepts are \((1,0)\) and \((5,0)\).

Step6: Plot the points and draw the parabola

Plot the vertex \((3,-4)\), the y - intercept \((0,5)\), the x - intercepts \((1,0)\) and \((5,0)\). Also, we can find a few more points by choosing values of \( x \) around the vertex. For example, when \( x = 2 \), \( y=(2 - 3)^2-4=1 - 4=-3 \); when \( x = 4 \), \( y=(4 - 3)^2-4=1 - 4=-3 \). Then, using the axis of symmetry and the direction of opening, draw a smooth upward - opening parabola passing through these points.

Answer:

The graph is a parabola with vertex at \((3, - 4)\), opening upwards, axis of symmetry \(x = 3\), y - intercept \((0,5)\), and x - intercepts \((1,0)\) and \((5,0)\). (To draw it, plot the vertex, intercepts, and additional points, then sketch the parabola.)