Exercise 18 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. 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. We can see that $a=1,$ $h=2,$ and $k=\text{-}1.$ 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 $(2,\text{-}1).$ 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 $x\text{-}$value and calculating its corresponding $y\text{-}$value. Let's try $x=4.$ When $x=4,$ we have $y=3.$ Thus, the point $(4,3)$ lies on the curve. Let's plot this point and reflect it across the axis of symmetry.

Note that both points have the same $y\text{-}$coordinate.

Step $4$

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