Mnemonics[ edit ] Equivalent to the right-triangle definitions, the trigonometric functions can also be defined in terms of the rise, run, and slope of a line segment relative to horizontal.

I woutould like to know how to find the equation of a quadratic function from its graph, including when it does not cut the x-axis. Modelling This is a good question because it goes to the heart of a lot of "real" math. Most "text book" math is the wrong way round - it gives you the function first and asks you to plug values into that function.

The parabola can either be in "legs up" or "legs down" orientation. We know that a quadratic equation will be in the form: Sometimes it is easy to spot the points where the curve passes through, but often we need to estimate the points.

Parabola cuts the graph in 2 places We can see on the graph that the roots of the quadratic are: But is this the correct answer? Here are some of them in green: So how do we find the correct quadratic function for our original question the one in blue?

System of Equations method To find the unique quadratic function for our blue parabola, we need to use 3 points on the curve. We can then form 3 equations in 3 unknowns and solve them to get the required result.

Vertex method Another way of going about this is to observe the vertex the "pointy end" of the parabola. We can write a parabola in "vertex form" as follows: If there are no other "nice" points where we can see the graph passing through, then we would have to use our estimate.

The next example shows how we can use the Vertex Method to find our quadratic function.

One point touching the x-axis This parabola touches the x-axis at 1, 0 only. What is the value of "a"?

But as in the previous case, we have an infinite number of parabolas passing through 1, 0. Here are some of them: We can see the vertex is at -2, 1 and the y-intercept is at 0, 2.

We just substitute as before into the vertex form of our quadratic function.Page 1 of 2 Translations and Reflections of Trigonometric Graphs Graphing a Horizontal Translation Graph y =2 cos 2 3 x º π 4.

SOLUTION Because the graph is a transformation of the graph of y =2cos 2 3 x, the amplitude is 2 and the period is = heartoftexashop.com comparing the given equation to the general equation. The graph of the cosine is the darker curve; note how it’s shifted to the left of the sine curve.

The graphs of y = sin x and y = cos x on the same axes. The graphs of the sine and cosine functions illustrate a property that exists for several pairings of the different trig functions. Equivalent to the right-triangle definitions, the trigonometric functions can also be defined in terms of the rise, run, and slope of a line segment relative to horizontal.

The slope is commonly taught as "rise over run" or rise / run. The three main trigonometric functions . Sine waves confused me. Yes, I can mumble "SOH CAH TOA" and draw lines within triangles.

But what does it mean?. I was stuck thinking sine had to be extracted from other shapes. Sample question: Write an equation of a sine curve with amplitude $\,5\,$, period $\,3\,$, and phase shift $\,2\,$.

Specifying a sine (or cosine) curve with a given amplitude, period, and phase shift defines a unique set of points in the plane. The choice of the values of x in the table correspond to x and y intercepts, maxima and minima points. These are useful points to graph the sine function over one period: [0, 2p].To graph f, we first graph the points in the table then join these points.

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Given Amplitude, Period, and Phase Shift, Write an Equation