Exercise 22 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=- 2x^2+8x+9 ⇔ y= - 2x^2+ 8x+ 9 We can see that a= - 2, b= 8, and c= 9. 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= - 2, b= 8

   x=- 8/2( - 2)

   ▼

   Simplify right-hand side

   MultPosNeg

   a(- b)=- a * b

   x=- 8/- 4

   DivNegNeg

   - a/- b=a/b

   x=8/4

   CalcQuot

   Calculate quotient

   x=2

   The axis of symmetry of the parabola is the vertical line with equation x=2.

   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=2. Therefore, the x-coordinate of the vertex is also 2. To find the y-coordinate, we need to substitute 2 for x in the given equation.

   y=- 2x^2+8x+9

   Substitute

   x= 2

   y=- 2( 2)^2+8( 2)+9

   ▼

   Simplify right-hand side

   CalcPow

   Calculate power

   y=- 2(4)+8(2)+9

   Multiply

   Multiply

   y=- 8+16+9

   AddSubTerms

   Add and subtract terms

   y=17

   We found the y-coordinate, and now we know that the vertex is (2,17).

   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, 9). 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= - 2, which is negative, the parabola will open downwards. Let's connect the three points with a smooth curve.