So how does the changing the number and a quadratic equation affect the graph? So if you have a positive, a value, what’s it’s going to cause either a stretch or a compression? ok, if you have a negative, a value, it will also cause a stretch or compression, but it will also cause a flip over your x axis if your a value is greater than one it’s going to cause a vertical stretch, meaning your a value causes your graph To get skinnier because our y value will increase quicker. If our a value is between 0 E 1, it will cause a vertical compression, meaning our graph will get wider, because our y value is increasing by a smaller amount. A positive c will shift up and a negative c will shift down. ok, the maximum and minimum of a quadratic function is also called the vertex. If you’re using a calculator to find your vertex, you would go second trace number three or number. Four then move the cursor to the left of the vertex press, enter and move the cursor to the right of the vertex press enter and then enter again. The other way you could potentially do it is by going menu, analyze and then maximum or minimum, and then using the same process of moving the cursor to one side, pushing enter moving the cursor to the other side and pushing enter again for you guys at home. You can use desmos to do that so we’ll put this equation into desmos, so three x, squared plus two x minus so y, equals three x squared and then was it plus 2, yeah plus 2x and then minus 1.
, 2x and then minus 1.. So our vertex is here it’s at the bottom it’s a minimum in this case, so negative, 0.33 or negative one third and then negative, one point three, three negative one and one third: what are the other words that mean the same thing as x, intercepts so x, Intercepts are also roots solutions zeros. Any of those words root solution. Zeros are just asking you: what are your x intercepts? So if we’re doing it in the calculator you put the equation in and then you’d find your x intercepts. If you guys are at home you’re using desmos, so you’ll, look at half and you’ll find your x intercepts, which are negative, 1, 0 E 0.3 0.. So how does the graph of y equals x squared minus 1 differ from the graph of y equals x, squared plus 7.? So this is the difference, the minus 1 and the positive 7. at the very end of our problem that will shift us up or down. Così, to get from negative one to positive seven, that means i added eight, so my graph shifted up eight units, so the graph of y equals x, squared minus one, is eight units to the left of the graph of x, squared plus seven that’s, not true right. We shifted up, we didn’t shift left or right. The graph of y equals x squared is eight units to the right of the graph of x, squared plus, seven again, also not true, because we shifted up not down or sorry up, not right.
The graph of x, squared minus one, is eight units above the graph of y equals x, squared plus seven plus seven is higher than minus one, so that one’s, not true the graph of x squared minus one, is eight units below the graph of x, squared plus Seven, that is true, and if we wanted to check this to make sure we could always put it into the calculator so x, squared minus one and i grade plus seven so i’ll type, that into desmos x, squared minus one y equals x, squared plus 7 and As we can see, the second one is higher than the first one x. Squared minus 1 È 8 units below x, squared plus, 7., ok, so the laws of exponents when multiplying the same base, we add the exponents when dividing the same base. We subtract our exponents. ok, so multiplying you get bigger, so we add our exponents together because addition makes us bigger when dividing you’re getting smaller. So we subtract our exponents because that makes us go smaller when you have a negative exponent, you’re going to turn it into a fraction and you’re going to flip it. So when you have a negative exponent blank to the other side of the fraction bar okay, so you’re going to flip it any base raised to the power of 0 is always equal to 1.. ok, so it wants to know which expression best represents. These three terms multiplied together: ok, quindi sono, always going to start with my coefficients.
My whole numbers that are in front of everything so 3 times negative three three times negative three is a negative, nine negative, nine times negative two would give me a positive eighteen, a squared times a well. If there is no exponents one and if we’re multiplying, we add our exponents together, so two plus one would be three. So i have a to the third or three a’s beats oop. I forgot about one of my a’s and then we also have a to the third again. So three plus three would make six. So we have a to the sixth b to the third times b. Well again, if there’s, no exponent, è, just one so three plus one is four times b again: there is no exponent, quindi è, one so b to the fourth plus b or four plus one would be five, so we have b to the fifth last one. We have is c times c to the third. Bene, if there is no exponent written it’s one and if we multiply, we add our exponents, so one plus three would be four, so we have c to the fourth. So our answer choice would be f all right, which expression best represents the simplification of the following. So again we always start with our coefficients. Our whole numbers out front 3 times negative 4 would be negative 12.. prossimo, we move on to our m’s so m to the negative 2 times m to the 6.
we’re going to add those two together, because we’re multiplying negative, two plus six would be positive four, so we have m to the positive four. Then we have n to the fourth times n to the negative seven. So four plus negative seven would be negative three, so we have n to the negative three. So since we have this negative exponent, it’s going to move to our denominator, so it’s just gon na shift down to the bottom, so we have negative 12 m to the fourth over in and then once we shift it. Our exponent will now become positive, so n to the third. So we end up with answer: Scelta, F, okay and then simplifying the last one, because it’s division i’m now going to subtract my exponents, ok, Così 5 diviso per 6. I cannot actually divide that out because i don’t want to end up with a decimal, so i want to try and simplify. If i can – and i can’t simplify that any further so it’s going to be 5 over 6., then we have a to the negative 3 and then a to the fifth okay, so it’s subtraction, so negative, 3 minus 5 would be a negative 8.. ora, since i’m, working with this in terms of already being in a fraction, a negative is just going to go to the bottom and it’s going to become positive so because it’s a to the negative 8 i’m going to write a to the 8 in my denominator.
Così, b to the fourth divided by b to the second, so four minus two would be two so we’re gon na go b squared at the top and then anything to the zero power is just one, so we’d technically be multiplying by one at the very bottom. But anything times, one is itself so this isn’t going to change.