Exercise 21 Page 61

To draw the graph of the given quadratic function, written in standard form, we must start by identifying the values of a, b, and c. y=x^2+2x+1 ⇔ y=1x^2+2x+1 We can see that a=1, b=2, and c=1. Now, we will follow four steps to graph the function.

1. Find the axis of symmetry.
2. Calculate the vertex.
3. Identify the y-intercept and its reflection across the axis of symmetry.
4. Connect the points with a parabola.

   Finding the Axis of Symmetry

   The axis of symmetry is a vertical line with equation x=- b2a. Since we already know the values of a and b, we can substitute them into the formula.
   x=- b/2a

   SubstituteII

   a= 1, b= 2

   x=- 2/2(1)

   ▼

   Simplify right-hand side

   Multiply

   Multiply

   x=- 2/2

   CalcQuot

   Calculate quotient

   x=- 1
   The axis of symmetry of the parabola is the vertical line with equation x=- 1.

   Calculating the Vertex

   Note that the formula for the x-coordinate of the vertex is the same as the formula for the axis of symmetry, which is x=- 1. To find the y-coordinate of the vertex, we need to substitute - 1 for x in the given equation.
   y=x^2+2x+1

   Substitute

   x= - 1

   y=( - 1)^2+2( - 1)+1

   ▼

   Simplify right-hand side

   CalcPow

   Calculate power

   y=1+2(- 1)+1

   Multiply

   Multiply

   y=1-2+1

   AddSubTerms

   Add and subtract terms

   y=0
   We found the y-coordinate, and now we know that the vertex is (- 1,0).

   Identifying the y-intercept and its Reflection

   The y-intercept of the graph of a quadratic function written in standard form is given by the value of c. Therefore, the point where our graph intercepts the y-axis is (0,1). Let's plot this point and its reflection across the axis of symmetry.

   

   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.