Exercise 10 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. h(x)=4(x+4)^2+6 ⇕ h(x)=4(x-(- 4))^2+6 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:& h(x)= 4(x-( - 4))^2+ 6 We can see that a= 4, h= - 4, and k= 6. 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 ( - 4, 6). Therefore, the axis of symmetry is the vertical line x= - 4.

   

   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=- 6.
   h(x)=4(x+4)^2+6

   Substitute

   x= - 6

   h( - 6)=4( - 6+4)^2+6

   ▼

   Simplify right-hand side

   SubTerm

   Subtract term

   h(- 6)=4(-2)^2+6

   CalcPow

   Calculate power

   h(- 6)=4(4)+6

   Multiply

   Multiply

   h(- 6)=16+6

   AddTerms

   Add terms

   h(- 6)=22
   When x=- 6, we have y=22. Thus, the point (- 6,22) 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!