Scaling and translating quadratic functions
We can get more general quadratic functions by scaling and translating the standard equation . Pleasantly, any quadratic function can be obtained in this way.
In this step we will see how scaling and translation affect the graph of a quadratic function.
Scaling the standard parabola
We have already seen that the shape of depends on the value of the constant . We can consider this graph to be a scaling of the particular standard parabola ; if then the parabola opens upwards, and if then the parabola opens downwards.
For large values of the parabola is steep and narrow. The closer to the value gets, the flatter the parabola. Indeed when the parabola is so flat that we get the straight line conic , which could be considered as a special case of the parabola.
Here for comparison are the graphs of (green), (blue) and (red).
Translating a standard parabola up or down
Now we investigate what happens if we translate a standard parabola up or down. This is quite easy. Note that however we translate a parabola, its vertex, and focus, and directrix will move in exactly the same way.
Q1 (E): Where is the vertex of the parabola obtained by translating by in the direction and by in the direction?
Geometrically, if we add a constant to the equation then we translate the parabola by in the vertical direction. The algebraic operation that shifts a function vertically by can be thought of as replacing with .
Recall that the parabola has vertex , focus , and directrix . Here is a graph of the parabola , obtained by translating down by . So its vertex will be at , its focus will be at , and its directrix will be the line .
Translating a standard parabola left or right
What if we want to shift over units to the right? We apply the same strategy as when we were translating lines: replace with to get
It is easy to check the old vertex has moved to .
We could expand the equation to get
Combining translations horizontal and vertical
If we take the standard parabola and translate it by in the direction and by in the direction, then we obtain
This is the general form of a standard parabola which has been translated by the vector, or directed line segment, . This is an important expression in the theory of quadratic functions.
Q2 (M): What is the vertex of this parabola?
Q3 (M): What are the focus and directrix of ?
Q4 (C): What is the focus of the parabola ?
A key fact about parabolas
Can we get every parabola this way – just by taking a standard parabola of the form , and then shifting or translating in the -direction by a certain amount and then in the -direction by a certain amount? Yes we can!
This is a pleasant and important fact about parabolas. It shows us that all quadratic functions, even ones with complicated formulas like have essentially similar shapes, and that we can find out where they are situated by unravelling how much and translations are required to obtain them from standard parabolas.
So there is a fundamental question here: how can we translate a standard parabola to get the general parabola ? Finding the answer will take us to ancient Persia, to a technique called completing the square, and to a remarkable identity that all students of mathematics ought to have seen.
A1. If we translate a parabola, then its vertex translates correspondingly. Since the vertex of is the origin , the vertex of the translated parabola must be .
A2. The unshifted parabola has a vertex at , so the new vertex will be at .
A3. The unshifted parabola has focus at and directrix . We have essentially just moved the entire picture to the right by and down by . So visually we can see the focus is now located at and the directrix is now .
A4. The focus of the parabola is the point .
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