One guiding principle of the SAT exam is to relate questions to the real world situations and the math section is no exception. This becomes rather complex when you are dealing with things like functions and complicated equations, but you need to become fluent in doing so.
The distance d traveled by an object starting with speed v with constant acceleration a over a period of time t is given by . Using this information, what is the expression for how much time it would take for an object starting from a standstill to travel a known distance d with acceleration a?
Here we are asked to solve for t in terms of the other variables. We’re told to assume we know the values of d and a and, since the object is starting from a standstill, its initial speed is v = 0. Then the equation is and we can solve for t by multiplying by 2, dividing by a, and taking the square root: .
The maximum range a projectile can be sent for a given initial speed is reached when launched at a 45° angle and is given by where d is range, v is initial speed and g is acceleration due to gravity. If the energy stored in an elastic band is completely transformed into kinetic energy, the distance the band is stretched is related to the projectile’s speed by = Energy = , where k is the spring constant, x is the distance stretched and m is the projectile’s mass. How would the spring constant need to be changed if you wanted to have the same range while pulling only half as far back on a slingshot using the same projectile?
First, we need to identify which variables we know and which we don’t. We’re asked how the spring k constant would change, so that’s the variable we’d like to solve for. On the other hand, using the same projectile and range means that we can treat the mass m and range d as known. We also know the distance stretched, since that’s just the original distance, divided by 2, and that gravity is essentially the same unless space travel is involved.
Let’s start by solving for the spring constant. We find . We know m and x but not v, so we need to check what we know about v from the other variables. In the formula , neither of the other variables (d and g) change, so v must not either. Now we’re ready to answer the question: When everything else remains the same, how does k change as a function of x? Specifically, if we replace x with , what happens to k? We can readily calculate this.
Calling the new spring constant :
Now was the original spring constant, so knew is four times the original spring constant. The answer, then, is that the spring constant must quadruple, or change by a factor of four.
We could also have solved for v2 and substituted this into the preceding expression to obtain an explicit formula for d in terms of the variables we’re told not to change (k, x, m and g). Then we would have found the relationship , which works just as well and is simply another perspective for the problem, which may help to make it more clear.
The logistic model is a famous model of population growth over time which was used to fit the following observations. When a small population is first introduced to a suitable environment, its growth rapidly accelerates as more individuals are born and more reproduce. Eventually, however, the ecosystem reaches its maximum carrying capacity due to factors such as limited food, and the growth slows as the population approaches a steady state.
Which of these graphs could represent a logistic function?
Which of these equations could represent a logistic function?
Note: Since x represents the amount of time that has passed, we only consider x ≥ 0.
Note that because we’re asked which of the functions could represent a logistic function and not given details about numeric values it should have, the numbers in the functions have limited importance. We only really need them to check significant characteristics, such as whether something is positive or negative, or increasing or decreasing.
To simplify this example, the graphs actually match up with the functions (graph 1 has the same shape as function 1 and so on), but a test question may only involve one and hence it’s important to understand both. Either way, we have the same starting point: translating what we’re told about the logistic function from words into mathematical behavior
First, what possible values can the logistic function have? Since the logistic function function’s output is the number of individuals in a population, it cannot be negative. Furthermore, if it’s zero at some time, it will remain at zero as time goes on since that means no individuals are present to reproduce. This removes graph number 4 from our possibilities. To see that the same applies for function 4, we could just plug in some convenient x-values and see what happens – for example, – or to be more thorough we could set f(x) = 0 and check to see whether the equation can hold true for some x ≥ 0.
Next, what general shape does it have? The population begins to increase rapidly at first but then its growth slows and approaches a steady value, so the slope should always be positive but flatten out as time goes on. Graph 1 flattens out as x increases, but it doesn’t fit the positive slope criterion – it has a peak and then decreases again as x increases. Likewise, function 1, is largest when its denominator is smallest. Since (x - 2)2 ≥ 0, the peak is at x = 2, and function 1 decreases for x smaller or larger than that.
Graph 3 has the opposite problem – though its slope is always positive, it continues to increase indefinitely and has no maximum. It’s easy to see that the same applies for function 3. That leaves us with only graph/ function 2, which fit all the criterion we inferred from the problem.
Of course, if you had a hunch and checked graph/function 2 first, you could stop as soon as you had confirmed your answer. If you have time, though, it doesn’t hurt to check the others – if another graph/function fits the requirements, it means you must have missed some of them when first reading the problem or perhaps that you made a mistake analyzing the options.
Some basic higher level math skills are being assessed by these questions, regardless of the item, concept or format. Being able to use these approaches when attempting to find the answers can help you toward a more successful experience.
Throughout the Passport to Advanced Math section, we’ve seen many examples of structure and how it can help us solve problems quickly – polynomials can be expanded or factored, the latter of which makes the polynomial’s zeroes easy to read; completing the square with a quadratic equation is a way of manipulating its structure so we can use a square root; graphs can be symmetric; and systems of equations be reorganized in ways that speed up computations. Recognizing certain structures in problems can give us lead on how to solve the problem.
It’s important to do your best to understand the ideas and thought processes used to solve mathematics problems rather than just knowing the procedure to solve certain types of problems. That way, if the problem is changed slightly from what you’ve worked with before, you can try to extend what you know to solve the new problem.
Understanding the ideas can also help you to zero in on the solution if it’s hidden in a complicated problem with extra information. For example, if you’re asked for values of y, the equation x3 + 2x2 - x + 11 = √(y - 10) has no real solutions. There’s no need to think about the large polynomial on the left side because the question asks about the values of y and, remembering that we can’t take the square root of a negative number, it’s immediately clear that we have no real solutions when y < 10.
The ability to relate a graph and a function is very useful, as a graph gives us a visualization of what a function does, and conversely a function contains all the information of the graph in a concise formula. Being comfortable with both representations gives you both an additional perspective on problems and flexibility in communicating information.