What is the range of x for positive values in f(x) = x^2 -2x -3?

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In summary, the function f(x) = x^2 - 2x - 3 has two zero crossing points at x = -1 and x = 3. The range of x for which the function is positive is x < -1 and x > 3. To graph the function without a graphic calculator, one can use a table of values and plot points on a set of axes. By choosing x values less than -1, between -1 and 3, and greater than 3, the positive regions of the function can be determined. Alternatively, a number line can also be used to visually track the behavior of the function.
  • #1
luigihs
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1. Plot the graph of thefunction f(x) = x^2 -2x -3 and determine the range of x for which the function is positive

x^2 -2x - 3
(x+1) (x-3) = 0
x = -1 and x = 3
Range -1, 3 <--- I am not sure am I right?


Cheers!
 
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  • #2
You found the zero crossing points, and there are two of them. One way to find out what the curve does in any region is to sketch it and see. Either use pencil and paper, or use one of the online graphing sites. Search google: online graphing
 
  • #3
NascentOxygen said:
You found the zero crossing points, and there are two of them. One way to find out what the curve does in any region is to sketch it and see. Either use pencil and paper, or use one of the online graphing sites. Search google: online graphing

Cheers! So the crossing points are x = -1 and x =3 , now I have to find how to graph in pencil because during the test I can't use graphic calculator ..
 
  • #4
Well you know what parabolas generally look like, and can determine whether they are shaped more like a u or an n by checking whether the coefficient (number in front) of the highest power (x2 in this case) is positive or negative (it can't be zero because then there would be no x2 term and it wouldn't be a parabola then). In this case the coefficient of x2 is 1, which is positive, so the parabola is shaped like a u.

Also keep in mind that a parabola is always symmetrical about its turning point (lowest or highest point) so since the zeroes are -1 and 3, so the parabola will have its lowest point directly in between the zeroes, at x=1.

And you can also plot points to help aid you! Try x=0 to find where it cuts the y-axis for example, and maybe even x=1 to know where the lowest point is located.
 
  • #5
From what i see: it'll be decreasing from ]-∞,1] and increasing from [1,+∞[
 
  • #6
Where the function is increasing of decreasing is not directly relevant to where the function is positive.


Better: [itex]x^2- 2x- 3= (x+ 1)(x- 3)[/itex] and the product of two number is positive only if the two numbers have the same sign. So: (x+1< 0 and x- 3< 0) or (x+1> 0 and (x- 3> 0). x+ 1< 0 for x< -1 and x- 3< 0 for x< 3. Those are both true for x< -1. x+1> 0 for x> -1 and x- 3> 0 for x> 3. Those are both true for x> 3.

More generally, any continuous function (and every polynomial is continuous) can change from "< 0" to "> 0" only where it "= 0". So we only need to check one point in each interval [itex](-\infty, -1)[/itex], [itex](-1, 3)[/itex], [itex](3, \infty)[/itex]. If x= -2, which is less than -1, [itex]f(-2)= (-2)^2- 2(-2)- 3= 4+ 4- 3= 5> 0[/itex] so f(x) is positive for all x< -3. If x= 0, which is between -1 and 3, [itex]f(0)= 0^2- 2(0)- 3= -3[/itex] so f(x) is negative for all x between -1 and 3. Finally, if x= 4, which is larger than 3, [itex]f(4)= 4^2- 2(4)- 3= 16- 8- 3= 5> 0[/itex].
 
  • #7
luigihs said:
Cheers! So the crossing points are x = -1 and x =3 , now I have to find how to graph in pencil because during the test I can't use graphic calculator ..
You draw two columns, heading one 'x' and the other 'y'. Divide the x interval -1 ... 3 into, say, 5, and write these x values in the x column. Calculate the y value for each of those x-values, knowing that y=x^2 -2x -3.

Finally, plot each x,y point on a set of axes.
 
  • #8
Since we know the graph intercepts the x-axis at x=-1 and x=3, we can pick x values that are less-than -1, in between -1 and 3, and greater then 3, and solve for each point. I like the points -2, 0, and 4. At x=-2 we get 5, at x=0 we get -3, at x=4 we get 5. With this data we clearly can determine where the function is positive. I usually prefer to do this using a number line so I can visually see and keep track of the functions behavior.
 
  • #9
i see...
 

1. What does it mean to determine the range of x?

When determining the range of x, you are trying to find the set of all possible values that x can take on in a given situation or equation.

2. How do you calculate the range of x?

The range of x can be calculated by finding the minimum and maximum values of x in a given set of data or by using mathematical equations or functions to determine the possible values of x.

3. Why is it important to determine the range of x?

Determining the range of x is important because it helps us understand the limitations or boundaries of a situation or equation. It also allows us to identify any possible outliers or extreme values that may affect our analysis.

4. Can the range of x be negative?

Yes, the range of x can be negative. It simply represents the set of all possible values that x can take on, whether they are positive, negative, or zero.

5. What factors can affect the range of x?

The range of x can be affected by various factors, such as the given data or equations, the number of data points, and any restrictions or limitations on the possible values of x.

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