Exercise 7 Page 575

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^2-6x ⇔ y=3x^2+(- 6)x+ We can see that a=3, b=- 6, and c= . 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= 3, b= - 6

   x=- - 6/2(3)

   ▼

   Simplify right-hand side

   Multiply

   Multiply

   x=- -6/6

   RemoveNegFracAndNum

   - - a/b= a/b

   x=6/6

   QuotOne

   a/a=1

   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/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=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=3x^2-6x

   Substitute

   x= 1

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

   ▼

   Simplify right-hand side

   CalcPow

   Calculate power

   y=3(1)-6(1)

   IdPropMult

   Identity Property of Multiplication

   y=3-6

   SubTerm

   Subtract term

   y=- 3
   We found the y-coordinate, and now we know that the vertex is (1, - 3).

   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, ). 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.