Graphs

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Quadratic Graphs 2

Some points on a quadratic curve are important to know and identify, such as the turning point, which we can find by completing the square.

There are a few key points on a curved graph that are important to identify. These act as the skeleton of the graph.

The turning point of a quadratic curve is the...

If we are given a quadratic function without a graph, we can deduce its turning point by rearranging the function and completing the square.

For example, let's find the turning point of the function $y = x^2 - 4x + 8$

Divide the coefficient of $x$ by 2 and place in square brackets with $x$

$(x - 2)^2$

Multiply out of the brackets

$(x - 2)^2 \space \rightarrow \space x^2-4x+4$

Find the difference

The original equation was $x^2-4x+8$ . We need to add on the difference between $(x-2)^2$ and the original equation.

Complete the square for $(x-2)^2$

The completed square is $(x - 2)^2+4$

This is an important step to finding the turning point

Now we can find the turning point

The turning point occurs when $(x - 2)^2=0$ . Therefore, we need to select a value of $x$ which makes this true.

Which value of $x$ is such that $(x-2)^2=0$ ?

When $x=2$ , $(x-2)^2=0$

Therefore, the x-coordinate of the turning point is $2$ .

To find the y-coordinate, substitute $x=2$ into the completed square

The completed square is $y = (x -2)^2 + 4$ .

If $y = (x -2)^2 + 4$ , what is $y$ when $x=2$ ?

The turning point is at $(2,4)$

Nice!

What is the minimum point of $y = (x - 3)^2 + 2$ ?