Quadratic equations part five

Quadratic equations Quadratic equations part five

You may have noticed that all of the quadratic graphs we have looked at are symmetrical and that they all have a U shape or an upside-down U shape. The bottom or top of the curve is called the turning point and, because the curve is symmetrical, the turning point is always halfway between the roots.

The key points on a graph of a quadratic equation are where the line of the graph crosses the y-axis, or the y-intercept, the roots, where the curve crosses the x-axis and the turning point which is the minimum or maximum point.

Let’s sketch $y = x^2 - x - 6$ by finding out these important features of the curve.

First, let’s factorise the equation

$(x-3)(x+2)$

From this factorisation, we can see that $y=0$ when $x=3$ and $x=-2$

What is the value of the y-intercept. The curve crosses the y-axis when $x=0$ so we just need to plug $y=0$ into $y = x^2 - x - 6$ which gives us $y = - 6$

Now we just need to calculate the coordinates of the turning point. The x-coordinate of the turning point is in the middle of the two points where the curve crosses the x-axis. The curve crosses the x-axis at $x=3$ and $x=-2$ and to find the middle of these numbers we can add them together and divide by two. So, the x-coordinate of the turning point is $\frac{3 -2}{2} = 0.5$ . To find the y-coordinate we just plug $x=0.5$ into = x^2 – x – 6$ to get = 0.5^2 – 0.5 – 6 = 0.25 – 1.5 – 6 = -6.25$

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