Graphing Quadratic Functions in Intercept Form (Factored Form)

Simple: Graph  $f(x)=-(x + 2)(x - 2)$

• Step 1: Identify x-intercepts: From the form  $f(x)=a(x-p)(x-q)$ , we have  $p=-2$  and  $q=2$ , so the x-intercepts are  $(-2,0)$  and  $(2,0)$ .
• Step 2: Find the axis of symmetry:  $x=\frac{-2 + 2}{2}=0$ .
• Step 3: Calculate the vertex: Substitute  $x = 0$  into the function:  $f(0)=-(0 + 2)(0 - 2)=-(2)(-2)=4$ , so the vertex is  $(0,4)$ .
• Step 4: Determine the direction of opening: Since  $a=-1<0$ , the parabola opens downward.
• Step 5: Plot points: Plot the x-intercepts  $(-2,0)$ ,  $(2,0)$  and the vertex  $(0,4)$ , then connect them to form the parabola.
• Key Features: X-intercepts  $(-2,0)$ ,  $(2,0)$ ; vertex  $(0,4)$ ; axis of symmetry  $x = 0$ ; domain  $(-\infty,\infty)$ ; range  $(-\infty,4]$ ; maximum value 4.

Medium: Graph  $f(x)=(x + 1)(x - 3)$

• Step 1: Identify x-intercepts:  $p=-1$ ,  $q = 3$ , so x-intercepts are  $(-1,0)$  and  $(3,0)$ .
• Step 2: Find the axis of symmetry:  $x=\frac{-1+3}{2}=1$ .
• Step 3: Calculate the vertex: Substitute  $x = 1$  into the function:  $f(1)=(1 + 1)(1 - 3)=(2)(-2)=-4$ , so the vertex is  $(1,-4)$ .
• Step 4: Determine the direction of opening: Since  $a = 1>0$ , the parabola opens upward.
• Step 5: Plot points: Plot the x-intercepts  $(-1,0)$ ,  $(3,0)$  and the vertex  $(1,-4)$ . To get more points, choose  $x=0$ :  $f(0)=(0 + 1)(0 - 3)=-3$ , so the point is  $(0,-3)$ ; its symmetric point about  $x = 1$  is  $(2,-3)$ . Plot these and connect to form the parabola.
• Key Features: X-intercepts  $(-1,0)$ ,  $(3,0)$ ; vertex  $(1,-4)$ ; axis of symmetry  $x = 1$ ; domain  $(-\infty,\infty)$ ; range  $[-4,\infty)$ ; minimum value -4.

Difficult: Graph  $f(x)=2(x + 4)(x + 1)$

• Step 1: Identify x-intercepts:  $p=-4$ ,  $q=-1$ , so x-intercepts are  $(-4,0)$  and  $(-1,0)$ .
• Step 2: Find the axis of symmetry:  $x=\frac{-4+(-1)}{2}=\frac{-5}{2}=-2.5$ .
• Step 3: Calculate the vertex: Substitute  $x=-2.5$  into the function:  $f(-2.5)=2(-2.5 + 4)(-2.5 + 1)=2(1.5)(-1.5)=2\times(-2.25)=-4.5$ , so the vertex is  $(-2.5,-4.5)$ .
• Step 4: Determine the direction of opening: Since  $a = 2>0$ , the parabola opens upward and is narrower than the parent function.
• Step 5: Plot points: Plot the x-intercepts  $(-4,0)$ ,  $(-1,0)$  and the vertex  $(-2.5,-4.5)$ . Choose  $x=-5$ :  $f(-5)=2(-5 + 4)(-5 + 1)=2(-1)(-4)=8$ , so the point is  $(-5,8)$ ; its symmetric point about  $x=-2.5$  is  $(0,8)$ . Plot these and connect to form the parabola.
• Key Features: X-intercepts  $(-4,0)$ ,  $(-1,0)$ ; vertex  $(-2.5,-4.5)$ ; axis of symmetry  $x=-2.5$ ; domain  $(-\infty,\infty)$ ; range  $[-4.5,\infty)$ ; minimum value -4.5.