(Real functions and equations) How to select points for a graph.

[Kyriakos1]

When I am given a function quadratic, square-root and inverse variation I am often uncertain as to how to select my points to graph the function. Usually I can find my vertex easily enough and y and x intercepts if any but otherwise I don't know how to select my points. Are there base points for each function? Such as (0,0), (1,1), (4,2), (8,2.8) for a function of square-root or (0,0), (1,1), (2,4), (-1,1), (-2,4) for a quadratic function.

 

[MarkFL]

If I am given a function of the form:

\(\displaystyle f(x)=a\sqrt{g(x)}+b\)

Then, I will find $x=x'$ such that:

\(\displaystyle g(x')=n^2\) where \(\displaystyle n\in\mathbb{N_0}\)

Then I plot the points:

\(\displaystyle (x,y)=(x',an+b)\)

If I am given a function of the form:

\(\displaystyle f(x)=a(x-h)^2+k\)

I let \(\displaystyle x=h+n\) where \(\displaystyle n\in\mathbb{N_0}\), and then for each point, reflect it across the axis of symmetry. You will get the set of points:

\(\displaystyle (x,y)=(h\pm n,an^2+k)\)

 

[Kyriakos1]

I thank you for answering and I do not mean to sound ungrateful but I don't really understand your explanation. I do not understand these symbols: n ∈ N 0 x′.
Also I have learned the square root-function as f(x)=a\sqrt{b(x-h)} + k and am unsure how to use f(x)=ag(x)−−−−√+b. Perhaps you can dumb it down a notch.

 

[MarkFL]

The statement \(\displaystyle n\in\mathbb{N_0}\) means that n is a natural number including zero, that is:

\(\displaystyle n\in\{0,1,2,3,\cdots\}\)

If you are given:

\(\displaystyle f(x)=a\sqrt{b(x-h)}+k\)

then set:

\(\displaystyle b(x-h)=n^2\implies x=\frac{n^2}{b}+h\)

which generates the points:

\(\displaystyle \left(\frac{n^2}{b}+h,an+k\right)\)

 

[CellCoree]

I understand the importance of accurately representing data through graphs and selecting appropriate points to plot can greatly impact the interpretation of the function. When graphing a quadratic function, it is important to remember that the shape of the graph is a parabola, which means it has a curved shape. Therefore, it is helpful to select points that are evenly spaced on both sides of the vertex. For example, if the vertex is at (0,0), you could select points such as (-2,4), (-1,1), (1,1), (2,4) to plot on the graph. This will give you a clear understanding of the shape of the parabola.

Similarly, for a square-root function, it is important to select points that are evenly spaced on both sides of the y-axis. This is because the square root function has a curved shape, similar to a parabola, but with a different rate of increase. So, for example, if the vertex is at (0,0), you could select points such as (-4,2), (-1,1), (1,1), (4,2) to plot on the graph.

For an inverse variation function, it is important to understand that the graph will have a hyperbolic shape. This means that the points selected should be evenly spaced on both sides of the asymptote (the line that the graph approaches but never touches). For example, if the asymptote is the x-axis, you could select points such as (-1,1), (-2,0.5), (1,1), (2,0.5) to plot on the graph.

In summary, there are no specific "base points" for each type of function, but it is important to select points that are evenly spaced and accurately represent the shape of the function. Additionally, it may be helpful to plot more points on the graph to get a better understanding of the behavior of the function.