Exercise 58 Page 329

y=-2(x-2)^2+3 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)= -2(x- 2)^2+ 3 We can see that a= -2, h= 2, and k= 3. 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 ( 2, 3). Therefore, the axis of symmetry is the vertical line x= 2.

   

   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=4.

   y=-2(x-2)^2+3

   Substitute

   x= 4

   y=-2( 4-2)^2+3

   ▼

   Simplify right-hand side

   SubTerm

   Subtract term

   y=-2(2)^2+3

   CalcPow

   Calculate power

   y=-2(4)+3

   MultNegPos

   (- a)b = - ab

   y=-8+3

   AddTerms

   Add terms

   y=-5

   When x=4, we have y=-5. Thus, the point (4,-5) 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!