Exercise 21 Page 557

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^2-20x ⇔ y=3x^2+(-20)x+ We can see that a=3, b=-20, 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= -20

   x=- -20/2(3)

   ▼

   Simplify right-hand side

   Multiply

   Multiply

   x=- -20/6

   RemoveNegFracAndNum

   - - a/b= a/b

   x=20/6

   ReduceFrac

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

   x=10/3

   The axis of symmetry of the parabola is the vertical line with equation x= 103, or x=3 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= 103. Thus, the x-coordinate of the vertex is also 103. To find the y-coordinate, we need to substitute 103 for x in the given equation.

   y=3x^2-20x

   Substitute

   x= 10/3

   y=3( 10/3)^2-20( 10/3)

   ▼

   Simplify right-hand side

   PowQuot

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

   y=3(100/9)-20(10/3)

   MoveLeftFacToNum

   a*b/c= a* b/c

   y=300/9-200/3

   ReduceFrac

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

   y=100/3-200/3

   SubFrac

   Subtract fractions

   y=-100/3

   The y-coordinate of the vertex is - 1003, or - 33 13. Therefore, the vertex is ( 103,- 1003), or (3 13,- 33 13).

   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.