Exercise 55 Page 110

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=3x^2+2x+1 We can see that a=3, 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.

   Let's do it!

   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= 3, b= 2

   x=- 2/2(3)

   ▼

   Evaluate right-hand side

   Multiply

   Multiply

   x=- 2/6

   ReduceFrac

   a/b=.a /2./.b /2.

   x=- 1/3
   The axis of symmetry of the parabola is the vertical line with equation x=- 13.

   Calculating the Vertex

   To calculate the vertex, we need to think of y as a function of x, y=f(x). We can write the expression for the vertex by stating the x- and y-coordinates in terms of a and b. Vertex: ( - b/2a, f( - b/2a ) ) Note that the formula for the x-coordinate is the same as the formula for the axis of symmetry, which is x=- 13. Therefore, the x-coordinate of the vertex is also - 13. To find the y-coordinate, we need to substitute - 13 for x in the given equation.
   y=3x^2+2x+1

   Substitute

   x= - 1/3

   y=3( - 1/3)^2+2( - 1/3)+1

   ▼

   Evaluate right-hand side

   NegBaseToPosPow

   (- a)^2 = a^2

   y=3(1/3)^2+2(- 1/3)+1

   PowQuot

   (a/b)^m=a^m/b^m

   y=3(1/9)+2(- 1/3)+1

   MultPosNeg

   a(- b)=- a * b

   y=3(1/9)-2(1/3)+1

   MoveLeftFacToNumOne

   a* 1/b= a/b

   y=3/9-2/3+1

   ExpandFrac

   a/b=a * 3/b * 3

   y=3/9-6/9+1

   OneToFrac

   Rewrite 1 as 9/9

   y=3/9-6/9+9/9

   AddSubFrac

   Add and subtract fractions

   y=6/9

   ReduceFrac

   a/b=.a /3./.b /3.

   y=2/3
   We found the y-coordinate and now we know that the vertex is (- 13, 23).

   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. Thus, 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=3, which is positive, the parabola will open upwards. Let's connect the three points with a smooth curve.

   

   We can see above that the minimum point of the curve is reached at the vertex.