Maths revision course by Miriam Hanks

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Maths revision course by Miriam Hanks Quadratics – What are they? x2 + 3x – 4 y2 – 4y + 2 1 – x2 Quadratics are equations where the highest power is 2. What shape do their graphs make? Maths revision course by Miriam Hanks

Maths revision course by Miriam Hanks Quadratic graphs Quadratics make a shape called a parabola It is a smiley face if the coefficient of x2 is positive, or a sad face if it’s negative. Maths revision course by Miriam Hanks

Maths revision course by Miriam Hanks Quadratics – Sketching graphs Any of these will help: Identify shape: or Find where it crosses the y-axis by putting x = 0 Find where it crosses the x-axis by putting y = 0 Complete the square to find the turning point Differentiate to find the turning point Maths revision course by Miriam Hanks

Maths revision course by Miriam Hanks Completing the square If there is a coefficient of x2, take it out as a factor of the first 2 terms. eg 2x2 + 12x – 5 = 2[x2 + 6] – 5 Now insert a new bracket, move the “squared” sign to the outside of it, halve the number inside, and square and subtract it: = 2[(x + 3)2 - 9] - 5 Maths revision course by Miriam Hanks

Maths revision course by Miriam Hanks Completing the square Next remove the outer bracket, remembering to multiply by your factor = 2(x + 3)2 - 18 – 5 Finally, tidy up the last two terms: = 2(x + 3)2 - 23 Maths revision course by Miriam Hanks

Maths revision course by Miriam Hanks Completing the square y =2(x + 3)2 - 23 How does this help draw the graph? The turning point is (-3, -23) Maths revision course by Miriam Hanks

Maths revision course by Miriam Hanks Solving a quadratic by using the formula. The quadratic formula is not given to you in the Higher maths exam, so you should learn it: Maths revision course by Miriam Hanks

Maths revision course by Miriam Hanks The discriminant = b2 – 4ac If b2 – 4ac = 0, then there is one real root (or two equal roots) Turning point is on x-axis If b2 – 4ac > 0, then there are 2 real roots Graph crosses x-axis twice If b2 – 4ac < 0, then there are no real roots Graph does not cross the x-axis Maths revision course by Miriam Hanks

Quadratic inequalities If your quadratic has a > or < sign, Start solving it as normal, but when you get the 2 solutions, decide on the direction of the < or > arrows by drawing a diagram. If you don’t draw the diagram, you will not get full marks in the exam. Maths revision course by Miriam Hanks

Quadratic inequalities example 1 Eg Solve x2 + 3x – 4 < 0 gives (x + 4) (x – 1) and so we mark x = -4 and x = 1 on the diagram: Since the original equation had a “< 0”, we look at where the graph is below the x-axis. Final answer: -4 < x < 1 -4 1 Maths revision course by Miriam Hanks

Quadratic inequalities example 2 Eg Solve x2 + 3x – 4 > 0 gives (x + 4) (x – 1) and so we mark x = -4 and x = 1 on the diagram: Since the original equation had a “> 0”, we look at where the graph is above the x-axis. Final answer: x < -4 and x > 1 -4 1 Maths revision course by Miriam Hanks

Maths revision course by Miriam Hanks Quadratics in real life Quadratics are used to make satellite dishes, suspension bridges and torches, all of which have a parabolic curve. Maths revision course by Miriam Hanks

Maths revision course by Miriam Hanks Quadratics in real life When you throw an object, the path it takes is also a parabola: Click for video clip Maths revision course by Miriam Hanks