How Can Jane Identify Graphs y1 and y3 Among the Functions Given?

[Gwozdzilla]

Homework Statement

Jane's calculus instructor asked her to draw the graphs of the functions y₁ = x^1/4f(x), y₂ = x^1/2f(x), y₃ = x²f(x), and y₄ = x⁴f(x) on the same coordinate plane using the same axes but different line-style for each graph to distinguish among them. Although he didn't specify f, identify y₁ and y₃ among the graphs Jane drew. I have attached the picture for this problem.

Homework Equations

None that I know of...

The Attempt at a Solution

I'm not really sure how to approach this problem, but I know that functions which are squared are generally shaped like upright or upsidedown U's, while functions that are raised to fractions like square roots or fourth roots are usually sideways U-shaped. However, they aren't asking about those specific functions but about those functions multiplied by another function. How do I know what the graph of an unknown function will look like when it's multiplied by a known function?
Thanks!

 

[pasmith]

Homework Statement

Jane's calculus instructor asked her to draw the graphs of the functions y₁ = x^1/4f(x), y₂ = x^1/2f(x), y₃ = x²f(x), and y₄ = x⁴f(x) on the same coordinate plane using the same axes but different line-style for each graph to distinguish among them. Although he didn't specify f, identify y₁ and y₃ among the graphs Jane drew. I have attached the picture for this problem.
View attachment 70032

Homework Equations

None that I know of...

The Attempt at a Solution

I'm not really sure how to approach this problem, but I know that functions which are squared are generally shaped like upright or upsidedown U's, while functions that are raised to fractions like square roots or fourth roots are usually sideways U-shaped. However, they aren't asking about those specific functions but about those functions multiplied by another function. How do I know what the graph of an unknown function will look like when it's multiplied by a known function?
Thanks!

Hint: It's clear from the graph that $f(x) > 0$ for $x > 0$, so multiplying by $f(x)$ preserves order.

 

[Gwozdzilla]

I'm sorry, but I'm not completely sure what you mean by "preserves order" or how that relates to the graphs. To me, all of the graphs look the same except for the one with large squares because that one is concave up. The rest of the functions are all concave down but they have different maximums. Do you mean that the graphs should still look similar to x^1/4 and x²? If that's the case, then the only graph that looks like x² to me is the one with large squares since it's concave up, but then there's no graph for x⁴, which would also be concave up. I guess you aren't referring to concavity then. I know that the order of a function is related to how many zeros are on its graph, but each of these graphs only has one visible zero and its the same for all of the functions, so I think I'm pretty confused... How is the order of a function related to its graph?

 

[pasmith]

I'm sorry, but I'm not completely sure what you mean by "preserves order"

A basic axiom of the real numbers is that for all real x and y and all strictly positive c, if $x \leq y$ then $cx \leq cy$. In other words, multiplication by a strictly positive constant preserves order. Thus, if $x^a < x^b$ for $0 < x < 1$ then, since $f(x) > 0$, it must follow that $x^af(x) < x^b f(x)$ for $0 < x < 1$.

This is all the information you need to work out which graph is which.

 

[Gwozdzilla]

Oh, I understand now. So the graphs remain in the same order as if they were multiplied by a positive constant, so the top solid line is y₁, the next one is y₂, then y₃, and y₄, because that's the order of the functions when a small fraction is put in for x. Therefore, the answer to the actual question is #4, the solid line for y₁ and the triangles for y₃. Is that correct?

Thanks!

 

[SammyS]

Oh, I understand now. So the graphs remain in the same order as if they were multiplied by a positive constant, so the top solid line is y₁, the next one is y₂, then y₃, and y₄, because that's the order of the functions when a small fraction is put in for x. Therefore, the answer to the actual question is #4, the solid line for y₁ and the triangles for y₃. Is that correct?

Thanks!

That's Correct !