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Samenvatting Wiskunde Wiskunde samenvatting Numbers And Space 3 vwo Mathematics summary Chapter one: Linear Relationships <ol> <li>Linear equations</li> </ol> <p style="margin-left:18.0pt">Solving an equation by ensuring that the variables only appear on the left-hand side: <p style="margin-left:18.0pt">10x-4=7x+20 <p style="margin-left:18.0pt">10x-7x=20+4 <p style="margin-left:18.0pt">When you move terms to the other side of the = sign, negative numbers become positive and positive numbers become negative. <p style="margin-left:18.0pt">How to solve linear equations: <ol> <li>Multiply out the brackets</li> <li>All terms containing x to the left-hand side and the rest to the right-hand side</li> <li>Simplify both sides</li> <li>Divide by the number in front of the x.</li> </ol> <ol> <li>Inequalities</li> </ol> <p style="margin-left:18.0pt">4(a-3) ≥ 4-3(5-a) linear inequality. This is a <p style="margin-left:18.0pt">4a-12 ≥ 4-15+3a <p style="margin-left:18.0pt">4a-3a≥4-15+12 <p style="margin-left:18.0pt">a ≥ 1 <p style="margin-left:18.0pt">When you divide by a negative number, the > and < symbols are flipped. <p style="margin-left:18.0pt">Solving an inequality works the same as solving a linear equation. Except that the last step could be to flip the < and > symbol. <p style="margin-left:18.0pt">X <sup style="line-height:1.5em">2 > 16 is a quadratic inequality. It results in x < -4 or x > 4. <p style="margin-left:18.0pt">The solutions to x2 < 16 lie between -4 and 4. <p style="margin-left:18.0pt">X lies between -4 and 4. <p style="margin-left:18.0pt">-4 < x < 4. <p style="margin-left:18.0pt">Leave square roots such as √2 as they are. <p style="margin-left:18.0pt">X <sup style="line-height:1.5em">2 < -16 no solutions <sup style="line-height:1.5em">2 > -16 any x has a solution. <p style="margin-left:18.0pt">X2 ≤ -16 no solutions -16 any x is a solution. X X2 ≥ <ol> <li>Linear formulas</li> </ol> <p style="margin-left:18.0pt">If there is a linear relationship between x and y, it will be in the form of y=ax+b. <ul> <li>The graph is a straight line.</li> <li>If you go 1 step to the right, you will go up a steps.</li> <li>The point of intersection with the y-axis is (0,b), so the y intercept is b.</li> </ul> When N=0.75t+1. The t-axis is the horizontal axis and the N-axis the vertical one. The graph intersects the N-axis (0,1). If you go 1 step to the right, you must go up 0.75 steps. Draw line <em style="line-height:1.5em">l</em>: y = -0.25x + 2. Point of intersection is A(0,2) on the y axis. Then use; <ul> <li> X = 4 results in y = -25 x 4 + 2 = 1. Therefore B(4,1).</li> <li>Or a = -0.25 means 1 to the right and 0.25 down. For example, 4 to the right and 1 down.</li> </ul> <p style="margin-left:36.0pt">How to generate a formula for a line: <p style="margin-left:36.0pt">You start with y=ax+b. b is the point of intersection with the y axis. Then select two coordinates of a grid point and divide them. <p style="margin-left:36.0pt">A = Vertical : Horizontal. <p style="margin-left:36.0pt">Lines <em style="line-height:1.5em">l</em> : y = 2x + 3 and <em style="line-height:1.5em">m</em> : y = 2x -8 are parallel because a is the same in both formulas. <p style="margin-left:36.0pt">For example: <p style="margin-left:36.0pt">Point A(4, -5) lies on line m : y = -3x + b. Calculate b. <p style="margin-left:36.0pt">How to work it out: <p style="margin-left:36.0pt">M : y = -3x + b <p style="margin-left:36.0pt">A (4, -5) on m. à -3 x 4 + b = -5. <p style="margin-left:36.0pt"> -12 + b = -5. <p style="margin-left:36.0pt"> b = -5 + 12. <p style="margin-left:36.0pt"> b = 7. <p style="margin-left:36.0pt">Generate the formula for line <em style="line-height:1.5em">l</em> which is parallel to line <em style="line-height:1.5em">m</em> : y = 5x – 1 and passes through point B(3,8). <p style="margin-left:36.0pt"><em style="line-height:1.5em">How to work it out:</em> <p style="margin-left:36.0pt">You know that l : y = ax+b. <p style="margin-left:36.0pt">l is parallel to m : y = 5x – 1, therefore a = 5. <p style="margin-left:36.0pt">The result is l : y = 5x + b <p style="margin-left:36.0pt">B(3,8) on l. à 5 x 3 + b = 8. <p style="margin-left:36.0pt"> 15 + b = 8. <p style="margin-left:36.0pt"> b = 8-15 <p style="margin-left:36.0pt"> b = -7. <p style="margin-left:36.0pt">Therefor l : y = 5x – 7. <ol> <li>Linear Functions</li> </ol> In 12 à 32, 12 is called the argument and 32 is the image. The arrow points from the argument to the image. Such a machine is called a function. 2x + 8 : x à 2x + 8. Another one: x à -2x + 6. For this function, the image 5 is equal to -2 x 5 + 6 = -10 + 6 = -4. Therefore 5 à -4. With functions, we call the argument x and the image y. So the function x à 2x + 5 means the same as the formula y = 2x + 5. Let’s name the function f. The image of 4 is equal to 2 x 4 + 5 = 13. f(4) = 13. Function <em style="line-height:1.5em">f </em>is given by x à 5x – 12. The function value of 3 is <em style="line-height:1.5em">f</em>(3) = 5 x 3 – 12 = 15 – 12 = 3. The function value of a random x is <em style="line-height:1.5em">f</em>(x) = 5x – 12. We call <em style="line-height:1.5em">f</em>(x) = 5x – 12 the brackets notation of <em style="line-height:1.5em">f.</em> Brackets notation: <em style="line-height:1.5em">f</em>(x) = 3x + 1. Y = 3x + 1. Functions such as <em style="line-height:1.5em">f</em>(x) = 3x – 1, <em style="line-height:1.5em">g</em>(x) = -x + 5 and <em style="line-height:1.5em">h</em>(x) = 5x are examples of linear functions. General form of a linear equation: <em style="line-height:1.5em">f</em>(x) = ax+b. For the graph of function f the following applies: x-intercept The y-coordinate is 0. The x-coordinate follows from f(x) = 0. The x-intercept is the solution to f(x) = 0. y-intercept The x-coordinate is 0. The y-coordinate is f(0) Therefore the y-intercept is f(0). The x-coordinate follows from f(x) = g(x). The y-coordinate is found by filling in the solution on f(x) or g(x). <ol> <li>Sum and difference graphs</li> </ol> When you add up 2 graphs, the new graph is called the sum graph. Then you can also draw the difference graph. You only need two points to draw a sum graph when the sum graph is a straight line. The sum graph of two lines is a straight line. When drawing it, you can use the points where each of the graphs intersect the x-axis. If you know the formulas of two graphs, you can easily work out the formula of the sum graph. If the formula for graph I is y = 0.5x + 1, and the formula for graph II is y = -x + 2, then the formula for the sum graph is y = 0.5x + 1 + -x + 2, or y = -0.5x + 3. There are 2 possibilities for the difference graph of graphs I: y = 0.5x + 1 and II : y = -x + 2. You can consider the difference graph I – II, but also the difference graph II – I: Y = 0.5x + 1 – (-x + 2), therefore y = 0.5x + 1 + x – 2, or y = 1.5x – 1. For difference graph II – I: Y = -x + 2 – (0.5x + 1), therefore y = -x + 2 – 0.5x – 1, or y = -1.5x – 1.