Exercise 9 Page 604

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=-1/2x^2+4x+1 ⇔ y=-1/2x^2+4x+1 We can see that a=- 12, b=4, 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= - 1/2, b= 4

   x=- 4/2(- 12)

   ▼

   Simplify right-hand side

   MultPosNeg

   a(- b)=- a * b

   x=- 4/- 1

   MoveNegDenomToFrac

   Put minus sign in front of fraction

   x=- ( - 4/1 )

   NegNeg

   - (- a)=a

   x=4/1

   DivByOne

   a/1=a

   x=4

   The axis of symmetry of the parabola is the vertical line with equation x=4.

   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=4. Thus, the x-coordinate of the vertex is also 4. To find the y-coordinate, we need to substitute 4 for x in the given equation.

   y=-1/2x^2+4x+1

   Substitute

   x= 4

   y=-1/2( 4)^2+4( 4)+1

   ▼

   Simplify right-hand side

   CalcPow

   Calculate power

   y=-1/2*16 +4(4)+1

   Multiply

   Multiply

   y=-8 +16+1

   AddTerms

   Add terms

   y=9

   We found the y-coordinate, and now we know that the vertex is (4,9).

   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=- 12, which is negative, the parabola will open downwards. Let's connect the three points with a smooth curve.