equivalent rational expressions are basically factoring in a fraction format and then once you have the expressions factored as I have in the second step of the expression and after that you must find the restriction where the denominator cannot be equal to 0 and the restrictions are -5 and 0
after that you cancel the like terms from the nominator and denominator which in this case is (x+5) and the final expression will be the one on the bottom
this week during math we learnt how to graph reciprocal linear and quadratic functions, in the picture above represented are 3 different ones. The first being linear the second being a simple quadratic function and the final being a more complex quadratic function.
the key to reciprocal functions is making sure you know exactly where your asymptotes are located and they will correspond with your interveniant numbers which are 1 and -1 and with your restrictions they represent where the hyperbolas will be located.
so for the first one x=3 and y=0 those will be our restrictions and the hyperbolas are set.
for the second one it’s more difficult because it’s a quadratic function but basically your asymptotes are what give you boundaries or sections and usually with a quadratic function it will be 6 sections as to which the parabola could be located in and simply which ever section the parabola is found in that’s where and which direction you draw the hyperbola the same goes for the bottom quadratic function
This week in math we learnt how to determine x-intercepts using the substitution method.
the very first step is you have to isolate a variable, in the equations above a variable has already been isolated for us which is “y” and from that step is the second where you substitute the letter variable which is “y” in the second equation to the equation above.
after that step 3 is distributing and and putting the equation in a form you can factor and then factor it to the point of being able to determine the x-intercepts and after that you do the math and the x-intercepts should be -5/4 and 1 for these equations
this week we learnt how to solve quadratic inequalities and how to test if the equation represented is true
above we have a standard quadratic equation but instead of being = to it is saying that the equation is greater than 0
so to test that we first have to find the x intercepts and from there we find a number lower than the lowest x intercept, a number in between the x intercepts, and a number greater than the greatest x intercept and we input them in the equation and see if the answer given is a positive or negative which will then tell us if the answer is greater or less than 0
we have now tested all three of the numbers we chose and it says that we have a positive, a negative, and another positive answer. We are looking for the positive answers to make this equation true.
after that you just state what is true which is….. X<-4 and x>2 making the equation true
the only way to find the x-intercepts is to bring it to factored form whether it be from standard form or general form and once you have it in factored form it’s simply like doing an equation between the brackets
The discriminant is related to the x-intercepts in the way that the discriminant will tell you how many intercepts you may have but in the factored form you figure out what the intercepts really are.
converting the equation from general form to standard form is what you must look to do when dealing with parabolas because standard form gives you just about all the information you need to form a parabola and all you need to do is complete the square just as I have done in the equation above
this week in math we learnt more about parabolas and I tried relating this week to last week and show you how helpful the discriminant truly is.
here we have 3 parabolas that show the different types of discriminant, in the first one the discriminant is greater than zero and intercepts the x-axis twice meaning we have 2 roots
in the next we have the parabola touch the x-axis which means we have only 1 real root
in the last one the parabola doesn’t even touch the x-axis which means we don’t have an intercept and no roots
It’s important to know the discriminant because then you know how many roots your equation will have, for example.
if discriminant < 0 then your equation will have 0 real roots
if discriminant = 0 then your equation will have exactly 1 root
if discriminant > 0 then your equation will have 2 roots
In the equation above I show you where to find the discriminant and I’ve done a question using it, in the question I did the discriminant is less than 0 meaning that it does not have any real roots
This week in math we learned about the quadratic formula
The use of it is basically to solve any quadratic equation and the way it works is….
the coefficient in front of the X squared is A, the coefficient in front of X is B, and finally the constant at the back will be C. you simply put the numbers into the formula after and do the math until you get an answer
This week in math we learned how to solve Radical Equations
up above we have one of the more complex radical equations but fairly easy if you know the steps to take
since they are both square roots you want to square them both to get rid of the root and from there you can begin the equation by moving the like terms to be paired together but don’t forget to switch the sign around once it crosses the equals sign, if it was positive it turns to negative and if it was negative it turns to positive. From there you combine the like terms which above gave us 20x and 10 and then you isolate the X by dividing 20 from both sides and you simplify to give you X=1/2. Now don’t forget to write down the restrictions, since the solution to X is greater than 0 we will put X is greater than or equal to 0