Exercise 12 Page 61

We want to draw the graph of the given quadratic function. Note that the function is already written in vertex form, f(x)=a(x-h)^2+k, where a, h, and k are either positive or negative numbers. y=1/2(x-3)^2+2 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:& y= 1/2(x- 3)^2+ 2 We can see that a= 12, h= 3, and k= 2. Since a is greater than 0, the parabola will open upwards.

   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 ( 3, 2). Therefore, the axis of symmetry is the vertical line x= 3.

   

   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.
   y=1/2(x-3)^2+2

   Substitute

   x= 5

   y=1/2( 5-3)^2+2

   ▼

   Simplify right-hand side

   SubTerm

   Subtract term

   y=1/2(2)^2+2

   CalcPow

   Calculate power

   y=1/2(4)+2

   Multiply

   Multiply

   y=2+2

   AddTerms

   Add terms

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