Exercise 53 Page 63

To graph the quadratic function given in intercept form f(x)=a(x-p)(x-q), we must start by identifying the values of a, p, and q. y=(x+3)(x-3) ⇕ y=1(x-(- 3))(x-3) We can see that a=1, p=- 3, and q=3. Now, we will follow four steps to graph the function.

1. Identify the x -intercepts.
2. Plot the x -intercepts and find the axis of symmetry.
3. Calculate the vertex and plot it.
4. Connect the points with a parabola.

   Identifying the x -intercepts

   Recall that the x-intercepts of a function written in intercept form are the values of p and q. Thus, the points where our function intercepts the x -axis are (- 3,0) and (3,0).

   Finding the Axis of Symmetry

   The axis of symmetry is halfway between the x -intercepts, (p,0) and (q,0). Therefore, it has the following equation. x= p+q2 Since we already know the values of p and q, we can substitute them into the formula.
   x=p+q/2

   SubstituteII

   p= - 3, q= 3

   x=- 3+3/2

   ▼

   Simplify right-hand side

   AddTerms

   Add terms

   x=0/2

   CalcQuot

   Calculate quotient

   x=0
   The axis of symmetry of the parabola is the vertical line with equation x=0. Let's plot the x -intercepts and the axis of symmetry on a coordinate plane.
   

   Calculating the Vertex

   The x -coordinate of the vertex is the same as the formula for the axis of symmetry, x= 0. To find the y -coordinate, we need to substitute 0 for x in the given equation.
   y=(x+3)(x-3)

   Substitute

   x= 0

   y=( 0+3)( 0-3)

   ▼

   Simplify right-hand side

   IdPropAdd

   Identity Property of Addition

   y=3(- 3)

   Multiply

   Multiply

   y= - 9
   We found the y -coordinate, and now we know that the vertex is (0,- 9). Let's plot it on our coordinate plane.
   

   Connecting the Points

   We can now draw the graph of the function. Since a=1, which is positive, the parabola will open upwards. Let's connect the three points with a smooth curve.