Exercise 3 Page 607

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=- 2x^2+10x-1 ⇕ y=- 2x^2+10x+(- 1) We can see that a=- 2, b=10, 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= 10

   x=- 10/2(- 2)

   ▼

   Simplify right-hand side

   MultPosNeg

   a(- b)=- a * b

   x=- 10/- 4

   RemoveNegFracAndDenom

   - a/- b= a/b

   x=10/4

   CalcQuot

   Calculate quotient

   x=2.5

   The axis of symmetry of the parabola is the vertical line with equation x=2.5.

   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=2.5. Thus, the x-coordinate of the vertex is also 2.5. To find the y-coordinate, we need to substitute 2.5 for x in the given equation.

   y=- 2x^2+10x-1

   Substitute

   x= 2.5

   y=- 2( 2.5)^2+10( 2.5)-1

   ▼

   Simplify right-hand side

   CalcPow

   Calculate power

   y=- 2(6.25)+10(2.5)-1

   MultNegPos

   (- a)b = - ab

   y = - 12.5+10(2.5)-1

   Multiply

   Multiply

   y=- 12.5+25-1

   AddSubTerms

   Add and subtract terms

   y=11.5

   We found the y-coordinate, and now we know that the vertex is (2.5,11.5).

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