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Section 4-2 : Parabolas

1. Sketch the graph of the following parabola. The graph should contain the vertex, the \(y\)‑intercept, \(x\)-intercepts (if any) and at least one point on either side of the vertex.

\[f\left( x \right) = {\left( {x + 4} \right)^2} - 3\]

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Start Solution

Let’s find the vertex first. Because the parabola is in the form \(f\left( x \right) = a{\left( {x - h} \right)^2} + k\) we know that the vertex is just the point \(\left( {h,k} \right)\). Therefore, we can compare our equation to this form and see that the vertex is : \(\left( { - 4, - 3} \right)\).

Be careful with minus signs here! For the \(h\) the term the general form is \({\left( {x - h} \right)^2}\) and so we need to write ours as \({\left( {x + 4} \right)^2} = {\left( {x - \left( { - 4} \right)} \right)^2}\) . This in turn means we must have \(h = - 4\) . Likewise, for the \(k\) in the general form it is “+\(k\)” and so to match our equation we need \(k = - 3\) .

Also note that \(a = 1 > 0\) for this parabola and so the parabola will open upwards.

Show Step 2

The \(y\)-intercept is just the point \(\left( {0,f\left( 0 \right)} \right)\). A quick function evaluation gives us that \(f\left( 0 \right) = 13\) and so for our equation the \(y\)-intercept is \(\left( {0,13} \right)\).

Show Step 3

For the \(x\)-intercepts we just need to solve the equation \(f\left( x \right) = 0\). So, let’s solve that for our equation.

\[\begin{align*}{\left( {x + 4} \right)^2} - 3 & = 0\\ {\left( {x + 4} \right)^2} & = 3\\ x + 4 & = \pm \sqrt 3 \\ x & = - 4 \pm \sqrt 3 \end{align*}\]

The two \(x\)-intercepts for this parabola are then : \(\left( { - 4 - \sqrt 3 ,0} \right) = \left( { - 5.7321,0} \right)\) and \(\left( { - 4 + \sqrt 3 ,0} \right) = \left( { - 2.2679,0} \right)\). Do not get excited about the “messy” values for the intercept coordinates. There is nothing wrong with them, they are just decimals rather than the integers we are used to dealing with.

Show Step 4

Because we had two \(x\)-intercepts for this parabola we already have at least one point on either side of the vertex and so we don’t really need to find any more points for our graph.

However, just for the practice let’s find the point corresponding to the \(y\)-intercept on the other side the vertex.

The \(y\)-intercept is a distance of 4 to the right of the vertex and so there will be a corresponding point at the same \(y\) value to the left and it will be a distance of 4 to the left of the vertex. Therefore, the point to the left of the vertex corresponding to the \(y\)-intercept is \(\left( { - 8,13} \right)\).

Show Step 5

Here is a sketch of the parabola including all the points we found above.