Exercise 23 Page 61

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=- 4x^2+8x+2 ⇔ y=- 4x^2+8x+2 We can see that a=- 4, b=8, and c=2. 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= - 4, b={\color{#009600}{\textcolor{#9400D3}8}}

   x=- 8/2(- 4)

   ▼

   Simplify right-hand side

   Multiply

   Multiply

   x=- 8/- 8

   RemoveNegFracAndDenom

   - a/- b= a/b

   x=8/8

   CalcQuot

   Calculate quotient

   x=1
   The axis of symmetry of the parabola is the vertical line with equation x=1.

   Calculating the Vertex

   Note that the formula for the x-coordinate of the vertex is the same as the formula for the axis of symmetry, which is x=1. To find the y-coordinate of the vertex, we need to substitute 1 for x in the given equation.
   y=- 4x^2+8x+2

   Substitute

   x= 1

   y=- 4( 1)^2+8( 1)+2

   ▼

   Simplify right-hand side

   CalcPow

   Calculate power

   y=-4(1)+8(1)+2

   Multiply

   Multiply

   y=-4+8+2

   AddTerms

   Add terms

   y=6
   We found the y-coordinate, and now we know that the vertex is (1,6).

   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 (0,2). 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=- 4, which is negative, the parabola will open downwards. Let's connect the three points with a smooth curve.