Exercise 24 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. f(x)=- x^2-6x+3 ⇔ f(x)= -1x^2+( -6)x+ 3 We can see that a= - 1, b= - 6, and c= 3. 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=- b2 a. Since we already know the values of a and b, we can substitute them into the formula.
   x=- b/2a

   SubstituteII

   a= - 1, b= - 6

   x=- - 6/2( - 1)

   ▼

   Simplify right-hand side

   Multiply

   Multiply

   x=- - 6/- 2

   RemoveNegFracAndDenom

   - a/- b= a/b

   x=- 6/2

   CalcQuot

   Calculate quotient

   x=- 3
   The axis of symmetry of the parabola is the vertical line with equation x=- 3.

   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/2 a, f( - b/2 a ) ) Note that the formula for the x-coordinate is the same as the formula for the axis of symmetry, which is x=-3. Thus, the x-coordinate of the vertex is also - 3. To find the y-coordinate, we need to substitute - 3 for x in the given equation.
   f(x)=- x^2-6x+3

   Substitute

   x= -3

   f( - 3)=- ( -3)^2-6( - 3)+3

   ▼

   Simplify right-hand side

   CalcPow

   Calculate power

   f(- 3)=- (9)-6(- 3)+3

   Multiply

   Multiply

   f(- 3)=-9+18+3

   AddTerms

   Add terms

   f(- 3)=12
   We found the y-coordinate, and now we know that the vertex is (-3,12).

   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, 3). 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.