Exercise 34 Page 199

We want to draw the graph of the given quadratic function. To do so, we will rewrite it in vertex form, f(x)=a(x-h)^2+k, where a, h, and k are either positive or negative numbers. f(x)=-(x-7)^2+10 ⇕ f(x)=-1(x-7)^2+10 To draw the graph, we will follow four steps.

1. Identify the constants a, h, and k.
2. Plot the vertex (h,k) and draw the axis of symmetry x=h.
3. Plot any point on the curve and its reflection across the axis of symmetry.
4. Sketch the curve.

   Let's get started.

   Step 1

   We will first identify the constants a, h, and k. Recall that if a<0, the parabola will open downwards. Conversely, if a>0, the parabola will open upwards. Vertex Form:& f(x)= a(x- h)^2+ k Function:& f(x)= -1(x- 7)^2+ 10 We can see that a= -1, h= 7, and k= 10. Since a is less than 0, the parabola will open downwards.

   Step 2

   Let's now plot the vertex ( h, k) and draw the axis of symmetry x= h. Since we already know the values of h and k, we know that the vertex is ( 7, 10). Therefore, the axis of symmetry is the vertical line x= 7.

   

   Step 3

   We will now plot a point on the curve by choosing an x-value and calculating its corresponding y-value. Let's try x=5.
   f(x)=-1(x-7)^2+10

   Substitute

   x= 5

   f( 5)=-1( 5-7)^2+10

   ▼

   Simplify right-hand side

   SubTerm

   Subtract term

   f(5)=-1(-2)^2+10

   CalcPow

   Calculate power

   f(5)=-1(4)+10

   MultNegPos

   (- a)b = - ab

   f(5)=-4+10

   AddTerms

   Add terms

   f(5)=6
   When x=5, we have y=6. Therefore, the point (5,6) lies on the curve. Let's plot this point and reflect it across the axis of symmetry.
   

   Note that both points have the same y-coordinate.

   Step 4

   Finally, we will sketch the parabola which passes through the three points. Remember not to use a straightedge for this!