Exercise 29 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=3/2x^2-3x+6 ⇔ y= 3/2x^2+( - 3)x+ 6 We can see that a= 32, b= - 3, and c= 6. 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=- b2 a. Since we already know the values of a and b, we can substitute them into the formula.
   x=- b/2a

   SubstituteII

   a= 32, b= - 3

   x=- - 3/2 ( 3/2)

   ▼

   Simplify right-hand side

   DenomMultFracToNumber

   2 * a/2= a

   x=- - 3/3

   RemoveNegFracAndNum

   - - a/b= a/b

   x=3/3

   CalcQuot

   Calculate quotient

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

   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/2 a, f( - b/2 a ) ) Note that the formula for the x-coordinate is the same as the formula for the axis of symmetry, which is x=1. Thus, the x-coordinate of the vertex is also 1. To find the y-coordinate, we need to substitute 1 for x in the given equation.
   y=3/2x^2-3x+6

   Substitute

   x= 1

   y=3/2( 1)^2-3( 1)+6

   ▼

   Simplify right-hand side

   CalcPow

   Calculate power

   y=3/2(1)-3(1)+6

   Multiply

   Multiply

   y=3/2-3+6

   AddSubTerms

   Add and subtract terms

   y=9/2

   CalcQuot

   Calculate quotient

   y=4.5
   We found the y-coordinate, and now we know that the vertex is (1,4.5).

   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, 6). 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= 32, which is positive, the parabola will open upwards. Let's connect the three points with a smooth curve.