Exercise 8 Page 61

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. g(x)=2(x+1)^2-3 ⇕ g(x)=2(x-(- 1))^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:& g(x)= 2(x-( - 1))^2+(- 3) We can see that a= 2, h= - 1, and k=- 3. 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 ( - 1,- 3). Therefore, the axis of symmetry is the vertical line x= - 1.

   

   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=- 3.
   g(x)=2(x+1)^2-3

   Substitute

   x= - 3

   g( - 3)=2(( - 3)+1)^2-3

   ▼

   Simplify right-hand side

   SubTerm

   Subtract term

   g(- 3)=2(- 2)^2-3

   CalcPow

   Calculate power

   g(- 3)=2(4)-3

   Multiply

   Multiply

   g(- 3)=8-3

   SubTerm

   Subtract term

   g(- 3)=5
   When x=- 3, we have y=5. Thus, the point (- 3,5) 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!