Exercise 30 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=- 5/2x^2-4x-1 ⇔ y= (- 5/2)x^2+(- 4)x+(- 1) We can see that a=- 52, 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= - 52, b= - 4

   x=- - 4/2 (- 52)

   ▼

   Simplify right-hand side

   DenomMultFracToNumber

   2 * a/2= a

   x=- - 4/- 5

   DivNegNeg

   - a/- b=a/b

   x=- 4/5

   CalcQuot

   Calculate quotient

   x=- 0.8
   The axis of symmetry of the parabola is the vertical line with equation x=- 0.8.

   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=1. Therefore, the x-coordinate of the vertex is also 1. To find the y-coordinate, we need to substitute 1 for x in the given equation.
   y=- 5/2x^2-4x-1

   Substitute

   x= 0.8

   y=- 5/2( - 0.8)^2-4( - 0.8)-1

   ▼

   Simplify right-hand side

   CalcPow

   Calculate power

   y=- 5/2(0.64)-4(- 0.8)-1

   Multiply

   Multiply

   y=- 1.6+3.2-1

   AddSubTerms

   Add and subtract terms

   y=0.6
   We found the y-coordinate, and now we know that the vertex is (- 0.8,0.6).

   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,- 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=- 52, which is negative, the parabola will open downwards. Let's connect the three points with a smooth curve.