Quadratic Functions and Parabolas

How does the graph of a non-linear function differ from that of a linear function?

Given the non-linear function $P = 25(1.016)^n$ to model Australia's future population, what does 'P' represent?

According to the model $P = 25(1.016)^n$, what is the estimated population of Australia in the year 2030?

If a quadratic function is expressed in the form $y = ax^2 + bx + c$, what term determines whether the parabola opens upwards or downwards?

What is the significance of the vertex of a parabola in the context of maximum and minimum problems?

What is the value of $y$ when $x$ is zero in the exponential function $y = b(a^x)$?

In the context of exponential functions, what does the term 'asymptote' refer to?

How does the graph of $y = a^x$ differ when $a > 1$ compared to when $0 < a < 1$?

What is the key feature of a reciprocal function that distinguishes it from linear and quadratic functions?

In a reciprocal function $y = k/x$, what happens to the value of $y$ as $x$ approaches infinity?

What term describes the relationship where one variable increases as the other decreases?

If $y$ varies directly as $x$, and $y = 10$ when $x = 2$, what is the constant of variation?

What type of symmetry does the graph of a simple reciprocal function $y = k/x$ exhibit?

Given that a quadratic function is represented by $y = ax^2 + bx + c$, how does changing the value of 'c' affect the graph?

The height of a ball thrown upwards is described by the quadratic function $h = -5t^2 + 20t + 25$. What does the constant term '25' represent within this context?

Ivy wants to enclose a lamb enclosure using a brick wall as one side and wire fencing for the other three sides. If she has 18 meters of fencing, how would you express the length $y$ in terms of the width $x$?

For the lamb enclosure described in the previous example, how can the area be written in terms of the width $x$?

How does the constant 'b' affect the graph of an exponential function $y = b(a^x)$?

How is the y-intercept of an exponential function commonly determined from its equation?

An exponential function is used to model the growth of a bacterial colony. If the function is given by $y = 500(1.2)^x$, where $y$ is the number of bacteria and $x$ is time in hours, what does the number 500 represent?

How does increased frequency of compounding affect an investment's exponential growth?

Why is declining-balance depreciation considered an example of exponential decay?

Given an exponential decay function $y = b(a^{-x})$, what happens to the value of $y$ as $x$ becomes increasingly large if $a > 1$?

What distinguishes a reciprocal function's graph from those of quadratic or exponential functions?

What is a practical implication of inverse variation as it relates to speed and time?

How does increasing the number of users typically affect Internet download speeds, assuming bandwidth is constant?

How do the axes relate to the graph of (y = k/x)?

What happens to the value of (y) in the equation (y = k/x) as (x) approaches zero?

If two variables exhibit direct variation, how does doubling one variable affect the other?

When a non-linear function models a real-world scenario, what does its graph often help to visualize?

How can you determine whether a parabola opens upwards or downwards by analyzing the quadratic equation $y = ax^2 + bx + c$?

How would you describe a parabola's “vertex,” and what significance does it hold in quadratic functions?

If given the exponential function $y = b(a^x)$ and you know that (a) is between 0 and 1, how does the graph behave as (x) increases?

What's one graphical characteristic that sets apart a reciprocal function (y = k/x) from linear, quadratic, and exponential functions?

Among the following cases, where is using a quadratic model potentially unsuitable for data when the time ( t ) is greater than 5?

Australia's population is being modeled in the form $P = 25(1.016)^n$, consider the influence of immigration and natural disasters.

Given the properties of each kind of function, where can a parabola be applicable, in terms of usage?

If you are going to plot the speed of vehicles and the accident counts at highways, what kind of graph are you approximately going to produce, with the presence of asymptotes?

You wish to accurately plot values given certain population data, but not exactly sure how to model, which one of these should be used?