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The following function, y = px3 ax? _ +rx +S; where and are the independent and dependent variables respectively, has horizontal tangents at the points (-2, 6) an...

Question

The following function, y = px3 ax? _ +rx +S; where and are the independent and dependent variables respectively, has horizontal tangents at the points (-2, 6) and (2, 0). Find the values of p,q, and [7 marks] Based on your answer in 2(a). find y' and state its domain and its range. [3 marks]

The following function, y = px3 ax? _ +rx +S; where and are the independent and dependent variables respectively, has horizontal tangents at the points (-2, 6) and (2, 0). Find the values of p,q, and [7 marks] Based on your answer in 2(a). find y' and state its domain and its range. [3 marks]



Answers

$$\text {Graph each horizontal parabola, and give the domain and range.}$$ $$-x=3 y^{2}+6 y+2$$

This is Paula. Number 15 access equal toe access. Tickle toe minus y Squire plus six Y plus seven. Okay, if you compare with X equal toe Ah e Y squad plus B Y plus C will be getting a call to minus one equal to minus one. Be equal to six and sequel to seven. Sequel to seven now Vertex will be X comma minus B by we end. One more thing is less than zero. So parabola will be left for opening like this. So except this is the Vertex so value of minds we were together. That is why a corner of the Vertex will be minus six by two and two minus one that is three. And for X coordinate Let us plug in the value I call to three in our original equation So x will be equal toe minus off Negative off three square plus six in +23 plus seven that is minus nine plus 18 plus seven that is 25 minus nine which is a call to 16. So Vertex will be equal toe 16 commentary. So this is our Vertex now. Ax intercept for X intercept. We will be plugging in. Why call to zero in our original equation? So X will be equal to zero plus seven. That is seven. So X intercept is seven comma zero. Okay, why intercept for Why intercept will plugging in Mexico to zero in our original question which is minus y squared plus six words plus 70 called aceto which can be written as by multiplying both sides by minus one. Why square minus six were minus seven equal to zero. So this is why square minus. Ah, we should write. Plus why minus seven, Why minus seven equals zero. Let's take virus common. So why plus one minus seven or a place on equal to zero. So why plus one? And why minus seven will be equal to zero. So we'll be having to values who I called tau minus one and seven. So there will be to intercept zero comma minus one and zero comma seven. These are the Y intercepts and acts intercept. We have seven common zero. So let us try and draw it. Vortexes 16. Common three. Okay, on the parabola is left for opening. So let us right here. 16 comma three. Next. Why? Let us take one in medical toe. Four for eight. 12. 16 for it. 12 16 20. Minus four. Minus eight. Minus 12. Minus 16. Minus 20. Minus four. Minus eight. 16. Commentary 16 comma three. So three will be somewhere here and 16 is yes. So 16 commentary is the Vertex axis of symmetry will be Oh, I called to three. So this is excessive. Cemetery what I call the three XX. This is X intercept is seven comma zero. So let us. Right. Seven common zero. Here. This is our ex Intercept seven comma zero. Why? Intercepts are zero comma minus 10 comma seven. So zero comma minus one will be somewhere here. This is zero comma, minus one and zero. Comma seven will be somewhere here. Zero comma seven. So our parabola should look like this. Okay, so if we talk about the domain, this is off course. The origin. If you talk about the domain domain will be minus infinity toe Last X value value of X for which the parable is possible. So 16 closed back at 16 and range will be off course minus infinity to plus infinity. Okay. These air dancers. Thank you

Yeah, number 18. Given a question is X equal toe minus y squared plus six way minus nine. Okay, so first of all, let us compare with X equal to a Y squired plus B Y plus C. We will be getting equal toe minus one and be equal to six. So over Tex is X comma minus B by to a which means why coordinate off the Vertex is minus me by two way, which is minus six by minus two. That is three. Yeah, And let us plug in this value of three in our original question to get the value fax. So x will be minus off. Three Squire plus six into three, minus nine. This will be minus nine plus 18 minus nine, which is equal to zero. So Vertex becomes Vertex becomes zero comma three. So this is our vertex. Okay, Now, toe, get ax intercept. We will be plugging in. Or why call to zero in our original question. So this will lead us to x equal toe minus nine. Okay, So minus nine comma zero is the X intercept for y intercept. We will be plugging in X equal to zero, so Ah, very question becomes minus y squared. Plus six Y minus nine. Equal to zero. Yeah, a little. Multiply both sides by minus one. This is why Squire minus six y plus nine equals zero, which is nothing. But why? Minus three Whole square. So off I will be equal to three. So why Intercept is a zero comma? Three. Okay, well intercepted. Zero commentary since, uh, why will have to Same value. That is why I called three and three. Because there is a whole square. So we can say that. Why? Excesses tendon shell toe the parabola. Okay, Now let us draw the parabola over Texas zero comma. Three acts interested to minus nine committed. Okay, So if this is why this is X So we have three six nine minus tree minus six, minus nine three six minus three minus six. Vertex is zero comma three. So this is the Vertex zero. Common. Three of the verdicts. Acts intercepted. Minus nine. Common zero. So this is our ex intercept minus nine comma zero. So axis of symmetry must be This is our axis of symmetry, which is what I call the three. So after a parable must look like this. It should pass through minus nine commands it also Okay? Yes. Supposed to passing through minus. We have to make it like this. Okay, got it. Okay, so in this case, our domain must be minus infinity to zero and the range minus infinity to plus infinity. Okay, so these are the values. Thank you.

Political behaviour work Bizarre problem number 70 here then creation off Para plays LINX minus y squared plus four way right. A six equals zero brewing thinkers y square minus four, right plastics because I see equals y minus true, the whole square plus two, two wrecks like us to Because why might a student who square Still we do X minus one? Because why it made us to the core square X minus? Because I want to do away mine us through the whole square where X is it but come back to But so already deferred Who are you in 50 and don't mind Isn't minus infinity to infinity.

We have. Why is it closed toe? One divided with three off excess choir minus six for X is a question My industry. Why is Equus toe minus three for X minus one. Why is minus 17 divided by 34 x Because to do you know why is minus six for excessive goes to one? Why is minus 17? Divided by three for X is equal to three. Why people took my industry. So let's plot this point on the graph. Andi Rather parable Lee Craft off wide because toe wonder. Well read three or fix a square Miller six. So this graph being drawn with the blue light. This is the required graph that passes Why exists from minus six. Now we will look for the domain and range of the function domain. It's clearly minus infinity to infinity that miss all real numbers. But from the graph, we object that the minimum all say the smallest value off. Why is Equus two minus six, So the rain will be including minus six to Infinity Exclusive


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