Question

graph the function.
$r(x) = (x + 1)^2$

Explanation:

Step1: Identify the vertex form

The function \( r(x) = (x + 1)^2 \) is in vertex form \( y = a(x - h)^2 + k \), where \( h = -1 \), \( k = 0 \), and \( a = 1 \). So the vertex is at \((-1, 0)\).

Step2: Find additional points

• When \( x = 0 \), \( r(0) = (0 + 1)^2 = 1 \), so the point is \((0, 1)\).
• When \( x = -2 \), \( r(-2) = (-2 + 1)^2 = 1 \), so the point is \((-2, 1)\).
• When \( x = 1 \), \( r(1) = (1 + 1)^2 = 4 \), so the point is \((1, 4)\).
• When \( x = -3 \), \( r(-3) = (-3 + 1)^2 = 4 \), so the point is \((-3, 4)\).

Step3: Plot the points and draw the parabola

Plot the vertex \((-1, 0)\), and the other points \((0, 1)\), \((-2, 1)\), \((1, 4)\), \((-3, 4)\). Since \( a = 1 > 0 \), the parabola opens upwards. Connect the points smoothly to form the graph of the quadratic function.

Answer:

The graph is a parabola opening upwards with vertex at \((-1, 0)\), passing through points like \((0, 1)\), \((-2, 1)\), \((1, 4)\), \((-3, 4)\) (plotted and connected smoothly).