Let's first recall the basic graphs as shown here in the diagram. The function's description is given below each graph. For example the linear function $$y=mx+b is shown on the left and the hyperbola $$y=1x is shown second from right.
Two of the graphs are shown with some generalising coefficients - the line $$y=mx+b and the exponential curve $$y=ax. Thus we could site particular examples of these graphs such as $$y=2x+5, $$y=7x and $$y=3x etc.
However, all of these basic types can be generalised further by what are known as transformations.
One common transformation is known as a translation. This is a horizontal and/or vertical shift in the position of the curve relative to the cartesian axes. Translations are easy to recognise.
If we replace $$x with $$(x−h), the basic curve is moved $$h units to the right. If we replace $$x with $$(x+h), the basic curve is moved $$h units to the left. If we replace $$y with $$(y−k), the basic curve is moved $$k units up. If we replace $$y with $$(y+k), the basic curve is moved $$k units down.
So suppose we take the hyperbola $$y=1x and translate it $$3 units to the right and $$5 units down. The new function becomes $$y+5=1x−3, or expressed explicitly $$y=1x−3−5. The vertical and horizontal asymptotes are respectively $$x=3 and $$y=−5.
If we take the parabola $$y=x2 and shift it $$1 unit to the left and $$5 units up, the new function becomes $$y−5=(x−1)2 . Again, we can express this explicitly as $$y=(x−1)2+5.
If we take the function $$y=x3 and translate it $$2 units to the right and $$1 unit up, the new function is given by $$y−1=(x−2)3, expressible as $$y=(x−2)3+1.
Another type of transformation is commonly referred to as a dilation. This is when a curve is stretched or compressed by some factor other than $$1 in the function's equation.
For example, the difference between $$y=x2 to $$y=3x2 is the dilation factor $$3. Every function value in $$y=3x2 is $$3 times the associated function value in $$y=x2. This means that the curve becomes steeper.
Again, every function value of the function $$y=2x is double the associated function value of $$y=1x.
Every function value of $$y=3(2x) is $$3 times that of $$y=2x, and every function value of $$y=12x3 is one-halve that of $$y=x3.
We can also reflect a curve across the $$x - axis by simply multiplying its function expression by $$−1.
Every function value of $$y=−x2 is the negative of those of $$y=x2, and the curve becomes an upside-down parabola with the same vertex as $$y=x2. Again, every function value of $$y=−3x is the reflected image of across the x -axis of $$y=3x.
When we combine translations and dilations we can reposition and distort the basic curve.
For example, the curve of $$y=5x−2+3 is constructed as follows. Start with the curve given by the basic function $$y=1x and stretch each ordinate by a factor of $$5. Then shift it $$2 units to the right and $$3 units up, so that the centre becomes $$(2,3) with the curve's asymptotes as $$x=2 and $$y=3.
The final graph is shown in red here, along with the graph of $$y=1x−2+3. Note how changing from $$y=1x−2+3 to $$y=5x−2+3 pushes the hyperbola out diagonally.
The $$y intercept of any function are found by putting $$x=0 into the equation. The $$x intercept(s) are found similarly by putting $$x=0.
For example, for the curve given by $$y=1x−3−5, putting $$x=0 reveals $$y=10−3−5=−513.
The x intercept is found by putting $$y=0, so that $$1x−3−5=0. This means that $$1x−3=5 and inverting $$x−3=15 so that $$x=315.
Consider the equation $$y=−2x.
Find the $$y-value of the $$y-intercept of the line.
Find the $$x-value of the $$x-intercept of the line.
Find the value of $$y when $$x=2.
Plot the equation of the line below.
Consider the function $$y=−12x2
Complete the following table of values.
$$x | $$−2 | $$−1 | $$0 | $$1 | $$2 |
---|---|---|---|---|---|
$$y | $$ | $$ | $$ | $$ | $$ |
Plot the graph.
Consider the cubic function $$y=4x3−3
Is the cubic increasing or decreasing from left to right?
Increasing
Decreasing
Is the cubic more or less steep than the function $$y=x3 ?
More steep
Less steep
What are the coordinates of the point of inflection of the function?
Inflection ($$, $$)
Plot the graph $$y=4x3−3
Consider the function $$y=−14x
Complete the following table of values.
$$x | $$−3 | $$−2 | $$−1 | $$1 | $$2 | $$3 |
---|---|---|---|---|---|---|
$$y | $$ | $$ | $$ | $$ | $$ | $$ |
Sketch the graph.
In which quadrants does the graph lie?
$$1
$$2
$$3
$$4