Exercise 4 Page 57

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. p(x)=- 2x^2-8x+1 ⇔ p(x)=- 2x^2+(- 8)x+1 We can see that a=- 2, b=- 8, 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= - 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=p(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, p( - 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.
   p(x)=- 2x^2-8x+1

   Substitute

   x= - 2

   p( -2)=- 2( -2)^2-8( - 2)+1

   ▼

   Simplify right-hand side

   CalcPow

   Calculate power

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

   Multiply

   Multiply

   p(- 2)=-8+16+1

   AddTerms

   Add terms

   p(- 2)=9
   We found the y-coordinate, and now we know that the vertex is (- 2,9).

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