Exercise 10 Page 604

To draw the graph of the given quadratic function written in standard form, let's start by identifying the values of a, b, and c. y=-2 x^2-3x+10 ⇔ y=-2x^2+(-3)x+10 We can see that a=-2, b=-3, and c=10. Now, we will follow three steps.

1. Find the axis of symmetry.
2. Calculate the vertex.
3. Graph the function.

   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= - 2, b= -3

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

   ▼

   Simplify right-hand side

   MultPosNeg

   a(- b)=- a * b

   x=- (-3/- 4)

   DivNegNeg

   - a/- b=a/b

   x=-3/4

   The axis of symmetry of the parabola is the vertical line with equation x=- 34.

   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=- 34. Thus, the x-coordinate of the vertex is also - 34. To find the y-coordinate, we need to substitute - 34 for x in the given equation.

   y=-2 x^2-3x+10

   Substitute

   x= -3/4

   y=-2 ( -3/4)^2-3( -3/4)+10

   ▼

   Simplify right-hand side

   PowQuot

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

   y=-2 (9/16)-3(-3/4)+10

   Multiply

   Multiply

   y=-18/16-3(-3/4)+10

   MultNegNegOnePar

   - a(- b)=a* b

   y=-18/16+9/4+10

   ExpandFrac

   a/b=a * 4/b * 4

   y=-18/16+36/16+10

   NumberToFrac

   a = 16* a/16

   y=-18/16+36/16+160/16

   AddFrac

   Add fractions

   y=178/16

   ReduceFrac

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

   y=89/8

   We found the y-coordinate, and now we know that the vertex is (- 34, 898). Rewriting the y-coordinate as a decimal, we can tell it is approximately 11.1.

   Graphing the function

   To graph the function we will make a table of values.

   x	- 2x^2-3x+10	y=- 2x^2-3x+10
   -3	- 2( -3)^2-3( -3)+10	1
   -2	- 2( -2)^2-3( -2)+10	8
   0	- 2( 0)^2-3( 0)+10	10
   1	- 2( 1)^2-3( 1)+10	5
   2	- 2( 2)^2-3( 2)+10	-4

   Now, let's plot the obtained points, then we can connect them with a parabola.