Sketching Quadratic Graphs 1
This chapter explores sketching quadratics. This is the first part. It covers sketching quadratics of the form f(x) = x2 + bx + c, sketching quadratics of the form f(x) = (x + p)2 + q and using completing the square to find the minimum point and factorising to find the intersects. Before attempting this chapter you must have prior knowledge of transforming simple graphs, and completing the sqaure.
Introduction to sketching quadratics
In previous lessons of plotting graphs we discovered how to draw graphs of y = x2 They always take the shape of a parabola. For example first we drew a table as shown below. We shall use the function y=x2 – 4
| -5 | 21 | -4 | 12 | -3 | 5 | -2 | 0 | -1 | -3 | 2 | 0 | 3 | 5 | 4 | 12 | 5 | 21 |
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We can also sketch quadratics without working out every coordinate. We use the skills we already have for transforming graphs and completing the square.
Investigating f(x) = x + bx + c
Here we shall explore the quadratic equation of the form;
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Below is the quadratic curve of f(x) = x2
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By changing b makes the graph go diagonally left or right. This also alters the min point and the roots as shwon below.
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C in the equation alters the intercept by moving it up and down as shown below.
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Investigating f(x) = (x + p) + q
In this section we shall explore quadratic in the fom of;
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Below is the quadratic curve of f(x) = x2 – 1
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Changing p translates the curve left or right p units as shown below.
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Changing q translates the curve up and donwn q units.
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How to sketch the curve of f(x) = (x + 2)-10
Below we’ll sketch the graph of f(x) = x2 graph translated. First we start off with the graph of f(x) = x2.
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The equation has 2 transformations the first one is;
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…moves the graph left by a units as shown below.
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…the next one is;
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This moves the graph up and down by a units as shown below.
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We shall carry out the transformations one at a time. Let us start with;
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We move the curve two places to the left this is caused by (x + 2)2 of the equation.
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Next we shall perform the second transformation, the second transform is;
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This will affect the height of the curve to move the graph up and down by a units. We move the curve by 10 units down because of the -10 in the equation.
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Above is the graph of;
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We can see that it has its roots around -5.2 and 1.2, and has a minimum point (-2, -10) with an intercept of y=-6.
We can also say that the curve has been translated by the following vector from f(x) = x2
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