Exercise 121 Page 298

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

   x=- 4/2(- 1)

   ▼

   Simplify right-hand side

   MultPosNeg

   a(- b)=- a * b

   x=- 4/- 2

   MoveNegDenomToFrac

   Put minus sign in front of fraction

   x=- ( - 4/2 )

   NegNeg

   - (- a)=a

   x=4/2

   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=- x^2+4x

   Substitute

   x= 2

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

   ▼

   Simplify right-hand side

   CalcPow

   Calculate power

   y=- 4+4(2)

   Multiply

   Multiply

   y=- 4+8

   AddTerms

   Add terms

   y=4
   We found the y-coordinate, and now we know that the vertex is (2,4).

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

   

   By looking at the graph we can state the values for the x-intercepts. We see that the parabola intercepts the x-axis at x=0 and x=4.