Graphing Functions

[jpanknin]

I'm working on graphing functions such as $$x^2+3x$$ with more than one x-value and really trying to understand how to "see" the graph from the equation/inequality before actually graphing it. I'm seeing some relationships, but wondering if there are any resources that can help. My book doesn't go into the intuition behind anything other than the basic function graphs (x^2, x^3, sqrt(x), x^2 - 3, (x+3)^2, etc.). But for equations like $$x^2+3x$$ it's a little more difficult.

I'm finding patterns such as for $$x^2+3x$$ the x-value of the vertex is at -1.5 (which is the negative midpoint of 0 and the coefficient 3 for x) and that the graph is below zero for all values -3 < x < 0. Likewise, for $$x^2+4x$$ the x-value of the vertex is -2 (which is the negative midpoint of 0 and the coefficient 4 for x) and the graph is below zero for all values -4 < x < 0.

I understand shifts like $$x^2+3$$ or $$(x+3)^2$$, but I'm having difficulty understanding the relationship/pattern when another x-variable is added and also when there are higher order terms as well like $$x^3+3x^2+2x+4$$. Any resources out there that discuss these well?

 

[JeffM]

This is a slightly weird but good question.

First, I suspect it is misworded. I think what is meant is a function in one variable that can be decomposed into more than one function in the same variable.

Second, it is psychologically odd. We look at a graph or even a sketch of a graph in order to supplement our intuition about a function. My intuition at least cannot cope unaided with $e^{x + \sin (x) }$. We look at a detailed graph to answer (usually approximately) numerical facts about a function. So asking our intuition to help make a graph seems to put the cart before the horse. And yet if our intuition is not well suited to guide graphing, how in the world can we construct a graph except by detailed numeric exploration?

Third, there is a mathematical tool that lets us sketch a graph easily for a huge class of functions, including virtually all functions treated in high school or beginning college. The same tool explains why a detailed graph behaves the way it does.That class of functions is called differentiable functions.

Here is the idea. Each differentiable function f(x) is uniquely related to a second function, often denoted by f’(x) and called the derivative of f(x), more precisely the first derivative of f(x). Most functions of interest in science or economics are twice differentiable. This just means that f’(x) also has a first derivative, which is denoted by f’’(x) and called the second derivative of f(x). The first and second derivatives give important and detailed information about f(x).

With respect to any functions f(x) and g(x) that are continuous in the interval (a, c) and differentiable in (a, b) and (b, c), we have the following rules

$$f’(x) > 0 \text { in } a, \ b) \iff f(x) \text { is everywhere increasing in } (a, \ b).\\ f’(x) > g’(x) > 0 \text { in } (a, \ b) \implies f(x) \text { is increasing faster than } g(x) \text { everywhere in } (a, \ b).\\ f’(x) < 0 \text { in } (a, \ b) \iff f(x) \text { is everywhere increasing in } (a, \ b).\\ f’(x) < g’(x) < 0 \text { in } (a, \ b) \implies f(x) \text { is decreasing faster than } g(x) \text { everywhere in } (a, \ b).\\ f’(x) \text { is not a constant} \text { in } (a, \ b) \iff f(x) \text { is curved rather than straight everywhere in } (a, \ b).\\ f‘(x) < 0 \text { in } (a, \ b), \ f‘(b) = 0, \text { and } f’x) > 0 \in (b, \ c) \implies f(b) \text { is a minimum, at least locally}.\\ f’(x) > 0 \text { in } (a, \ b), \ f’(b) = 0, \text { and } f’(x) < 0 \in (b, \ c) \implies f(b) \text { is a maximum, at least locally}.$$
If f(x) is twice differentiable, we get additional information.

The study of what functions are differentiable, how to find the derivatives of various functions, why the rules given above make sense, and how to derive practical results from those rules is what is studied in differential calculus. Any text on differential calculus will have a chapter or section on sketching graphs (it may be under the topic of “critical points”).

 

[Steven G]

1st you really should name your function. y or f(x) would work in your example.

f(x)=x^2 + 3x.

I would let g(x) = x^2 and h(x) = 3x. Then f(x) = g(x) + h(x).

Now graph both g(x) and h(x) on the same x-y coordinate system. Then for random x-values add the two corresponding y-values to find the f(x) for these x-values. You should start seeing how the graph of f(x) looks like.

I think a better example for you to look at would be f(x) = 1/x +x for x>0

 

[jpanknin]

This is a slightly weird but good question.

First, I suspect it is misworded. I think what is meant is a function in one variable that can be decomposed into more than one function in the same variable.

Second, it is psychologically odd. We look at a graph or even a sketch of a graph in order to supplement our intuition about a function. My intuition at least cannot cope unaided with $e^{x + \sin (x) }$. We look at a detailed graph to answer (usually approximately) numerical facts about a function. So asking our intuition to help make a graph seems to put the cart before the horse. And yet if our intuition is not well suited to guide graphing, how in the world can we construct a graph except by detailed numeric exploration?

Third, there is a mathematical tool that lets us sketch a graph easily for a huge class of functions, including virtually all functions treated in high school or beginning college. The same tool explains why a detailed graph behaves the way it does.That class of functions is called differentiable functions.

Here is the idea. Each differentiable function f(x) is uniquely related to a second function, often denoted by f’(x) and called the derivative of f(x), more precisely the first derivative of f(x). Most functions of interest in science or economics are twice differentiable. This just means that f’(x) also has a first derivative, which is denoted by f’’(x) and called the second derivative of f(x). The first and second derivatives give important and detailed information about f(x).

With respect to any functions f(x) and g(x) that are continuous in the interval (a, c) and differentiable in (a, b) and (b, c), we have the following rules

$$f’(x) > 0 \text { in } a, \ b) \iff f(x) \text { is everywhere increasing in } (a, \ b).\\ f’(x) > g’(x) > 0 \text { in } (a, \ b) \implies f(x) \text { is increasing faster than } g(x) \text { everywhere in } (a, \ b).\\ f’(x) < 0 \text { in } (a, \ b) \iff f(x) \text { is everywhere increasing in } (a, \ b).\\ f’(x) < g’(x) < 0 \text { in } (a, \ b) \implies f(x) \text { is decreasing faster than } g(x) \text { everywhere in } (a, \ b).\\ f’(x) \text { is not a constant} \text { in } (a, \ b) \iff f(x) \text { is curved rather than straight everywhere in } (a, \ b).\\ f‘(x) < 0 \text { in } (a, \ b), \ f‘(b) = 0, \text { and } f’x) > 0 \in (b, \ c) \implies f(b) \text { is a minimum, at least locally}.\\ f’(x) > 0 \text { in } (a, \ b), \ f’(b) = 0, \text { and } f’(x) < 0 \in (b, \ c) \implies f(b) \text { is a maximum, at least locally}.$$
If f(x) is twice differentiable, we get additional information.

The study of what functions are differentiable, how to find the derivatives of various functions, why the rules given above make sense, and how to derive practical results from those rules is what is studied in differential calculus. Any text on differential calculus will have a chapter or section on sketching graphs (it may be under the topic of “critical points”).

I'm slightly weird, @JeffM . So that makes sense . I've taken calculus and now I'm revisiting the fundamentals of algebra because I don't feel like I really learned these concepts. More just memorized them for exams and didn't really understand them. So as I walk through the process I often come up with questions that help me get a more intuitive feel for what the math is "saying" and I have to say it's made math a lot more understandable and a lot more fun.

 

[jpanknin]

1st you really should name your function. y or f(x) would work in your example.

f(x)=x^2 + 3x.

I would let g(x) = x^2 and h(x) = 3x. Then f(x) = g(x) + h(x).

Now graph both g(x) and h(x) on the same x-y coordinate system. Then for random x-values add the two corresponding y-values to find the f(x) for these x-values. You should start seeing how the graph of f(x) looks like.

I think a better example for you to look at would be f(x) = 1/x +x for x>0

This is a good approach and definitely makes sense. Thanks @Steven G.

 

[Dr.Peterson]

I'm working on graphing functions such as $$x^2+3x$$ with more than one x-value and really trying to understand how to "see" the graph from the equation/inequality before actually graphing it. I'm seeing some relationships, but wondering if there are any resources that can help. My book doesn't go into the intuition behind anything other than the basic function graphs (x^2, x^3, sqrt(x), x^2 - 3, (x+3)^2, etc.). But for equations like $$x^2+3x$$ it's a little more difficult.

I'm finding patterns such as for $$x^2+3x$$ the x-value of the vertex is at -1.5 (which is the negative midpoint of 0 and the coefficient 3 for x) and that the graph is below zero for all values -3 < x < 0. Likewise, for $$x^2+4x$$ the x-value of the vertex is -2 (which is the negative midpoint of 0 and the coefficient 4 for x) and the graph is below zero for all values -4 < x < 0.

I understand shifts like $$x^2+3$$ or $$(x+3)^2$$, but I'm having difficulty understanding the relationship/pattern when another x-variable is added and also when there are higher order terms as well like $$x^3+3x^2+2x+4$$. Any resources out there that discuss these well?

I don't know that there is a general way to recognize what a function will look like at a glance; each type of function has its own issues.

In this case, the first thing I would do is to factor it and find the x-intercepts, in addition to recognizing it as a parabola. (The vertex is halfway between the x-intercepts.) I might also complete the square, or just use the fact that the line of symmetry is x = -b/(2a).

 

[Steven G]

I have to say it's made math a lot more understandable and a lot more fun.

Yes, math is fun if you understood it rather than memorize it!

 

[JeffM]

I'm slightly weird, @JeffM . So that makes sense . I've taken calculus and now I'm revisiting the fundamentals of algebra because I don't feel like I really learned these concepts. More just memorized them for exams and didn't really understand them. So as I walk through the process I often come up with questions that help me get a more intuitive feel for what the math is "saying" and I have to say it's made math a lot more understandable and a lot more fun.

Because you have studied calculus, you can use it (and geometry if you remember any of it) to refine your understanding of algebra.

Technically, calculus strictly defined is dependent on algebra strictly defined, and so that is the way they are taught (algebra first). But a lot of what is introduced in "algebra" courses covers more than algebra strictly defined. When you are going back to refine your understanding, there is no reason to limit yourself to what a novice knows.