c1: chapters 1-4 revision
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C1: Chapters 1-4 Revision. Dr J Frost ([email protected]) . Last modified: 10 th October 2013. Solving simultaneous equations. Remember that the strategy is to substitute the linear equation into the quadratic one, then solve. ?. Expanding out correctly!. ?. Inequalities. - PowerPoint PPT PresentationTRANSCRIPT
Solving simultaneous equationsRemember that the strategy is to substitute the linear equation into the quadratic one, then solve.
𝑥+ 𝑦=2
𝑥=13 , 𝑦=
53∨𝑥=5 , 𝑦=−3?
Expanding out correctly!
1−2 (𝑥−2 )2?
Find the set of values of x for which(a) 4x – 3 > 7 – x (b) 2x2 – 5x – 12 < 0 (c) both 4x – 3 > 7 – x and 2x2 – 5x – 12 < 0
Inequalities
𝑥>2
− 32 <𝑥<4
2<𝑥<4
Remember for quadratic inequalities:1. Always start by putting in the form or .
If you have , ABSOLUTELY DON’T divide by , but write 2. Then factorise.3. Then sketch.
Your answer will either be , or “ or ”. Be sure to use the word ‘or’ in the latter one, since ‘and’ would be wrong.
For inequalities in general:• Multiplying/dividing both sides by a negative number flips the inequality.• Don’t mix up AND and OR. “” is different from “”.
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The equation , where k is a constant, has 2 different real solutions for x.(a) Show that k satisfies
(b) Hence find the set of possible values of k.
DiscriminantWhenever you see the words “equal roots”, “distinct/different roots” or “no roots”, you know you’ve got to calculate the discriminant, which is .• It helps to explicitly write out your , and first before substituting into the
discriminant.• Be VERY careful with double (or even triple!) negatives.
The discriminant of is .The discriminant of is 4.
• When you have ‘different roots’ or ‘no roots’, you’ll have a quadratic inequality. Solve in the same way as before. But remember your sketch is in terms of , not in terms of the original variable . So don’t be upset if your sketch has roots, even if the original question asks where your equation has no roots.
𝑘<1𝑜𝑟 𝑘>4?
??
Sketching quadratics/cubics
• For cubics, think whether the / term is positive or negative. Cubics with positive will go uphill, and downhill otherwise.
• If , without fully expanding you can tell you’ll have a term, thus it goes downhill. Be careful though: in , the term will be positive!
• You can get the roots/-intercepts by setting to be 0. Imagine each factor/brackets being 0. So if , then the roots are
• For both quadratics and cubics, the curve touches the x-axis for a root if the factor is squared, and crosses if not repeated.
• Don’t forget the y-intercept! YOU WILL LOSE MARK(S) OTHERWISE.• It’s quite acceptable to have algebraic expressions as roots/y-intercepts. The y-
intercept is ? No problem!
• Don’t forget what a sketch of or looks like.
Sketching cubicsSketch the following, ensuring you indicate the values where the line intercepts the axes.
y = (x+2)(x-1)(x-3)
y = x(x-1)(2-x)
y = x(2x – 1)(x + 3)
y = x2(x + 1)
y = x(x+1)2
y = x(1 – x)2
y = -x3
y = (x+2)3
y = (3-x)31
2
3
4
5
6
7
8
9
10
11
12
y = (x+2)2(x-1)
y = (2-x)(x+3)2
y = (1 – x)2(3 – x)
-2 1 3
6
1 2
0.5 3
-1
-1
1
-2
8
1
3
-3
18
-2 1
3
27
-4
2
3
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Transforming Existing Graphs
a f(bx + c) + dBro Tip: To get the order of transformations correct inside the f(..), think what you’d need to do to get from (bx + c) back to x.
Step 1: c
Step 2: ↔ b
Step 3: ↕ a
Step 4: ↑ d
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Transforming Existing GraphsHere is the graph y = f(x). Draw the following graphs, ensuring you indicate where the graph crosses the coordinate axis, minimum/maximum points, and the equations of any asymptotes.
(2, 3)
1
x
y
y = -1
y = f(x)
6
x
y
y = -2
y = 2f(x+2)
y = f(2x)
1
x
y
y = -1
(1, 3)
y = -f(-x) – 1
-2
x
yy = 0
(-2, -4)
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Bro Tip: Don’t get to transform the asymptotes! This horizontal asymptote won’t be affected by any transformations, but will by ones.
Sketching Graphs by Considering the Transform
It’s often helpful to consider a simpler graph first, e.g. or , and then consider what transform we’ve done.
Sketch
Start with Then clearly we’ve replaced with and added 4 to the result.
i.e. ??
Sketch
x
y
-2
𝑦=−1
𝑥=−1
-0.5
?
Sketching Graphs by Considering the Transform
Sketch
x
y
12
𝑥=2?
Sketching Graphs by Considering the Transform
Sketching Quadratics
Sketch y = x2 + 2x + 1 Sketch y = x2 + x – 2
x
y
x
y
1
1-2 1
-2
Sketch y = -x2 + 2x + 3
x
y
3
-1 3
Sketch y = 2x2 – 5x – 3
x
y
-3-0.5 3
? ?
? ?
Sketching Quadratics
Sketch y = x2 – 4x + 5
x
y
(2, 1)5
Sketch y = -x2 + 2x – 3 y
-3(1,-2)
? ?
Some quadratics have no roots. In which case, you’ll have to complete the square in order to sketch them. This tells you the minimum/maximum point.
e.g. So minimum point is .-intercept is 3
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