Exercise 9 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=- 0.5x^2+2x+1 We can see that a=- 0.5, b=2, and c=1. 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= - 0.5, b= 2

   x=- 2/2(- 0.5)

   ▼

   Simplify right-hand side

   MultPosNeg

   a(- b)=- a * b

   x=- 2/- 1

   RemoveNegFracAndDenom

   - a/- b= a/b

   x=2/1

   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/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=2. Thus, 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=- 0.5x^2+2x+1

   Substitute

   x= 2

   y=- 0.5( 2)^2+2( 2)+1

   ▼

   Simplify right-hand side

   CalcPow

   Calculate power

   y=- 0.5(4)+2(2)+1

   Multiply

   Multiply

   y=- 2+4+1

   AddSubTerms

   Add and subtract terms

   y=3
   We found the y-coordinate, and now we know that the vertex is (2,3).

   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 (0,1). 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=- 0.5, which is negative, the parabola will open downwards. Let's connect the three points with a smooth curve.