Mastering Linear Systems and Quadratic Functions

Finding the Point of Intersection

Method 1: Elimination

1. Use multiplication to get one variable having the same numbers in both equations (they can have different signs).
2. If the signs for the variables are the same, subtract the two equations; otherwise, add them.
3. Substitute the answer into an original equation and solve.

Method 2: Substitution

1. Isolate one variable in one equation.
2. Substitute into the other equation.
3. Solve.
4. Substitute the answer into the original and solve.

Triangle Centers and Properties

Finding the Median

1. Determine the midpoint of the opposite line using the midpoint formula.
2. You now have two points; determine the slope of the median.
3. Use the slope and one point to determine the equation of the median.

Finding the Altitude

1. Determine the slope of the opposite line.
2. Determine the perpendicular slope.
3. Use the perpendicular slope and point to determine the equation.

Finding the Perpendicular Bisector

1. Determine the slope of the given line.
2. Determine the perpendicular slope.
3. Use the perpendicular slope and point on the given line to find the equation.

Classifying Triangle Properties

• Equilateral Triangles: Have all the same side lengths.
• Isosceles Triangles: Have two equal sides.
• Right Triangles: Have one right angle or lines that are perpendicular.

Quadratic Relations and Factoring

In quadratic relations, if a is larger than 1 or lower than -1, the graph is stretched; otherwise, it is compressed.

Factoring Quadratic Equations

• Technique 1: Common factoring (looking for the Greatest Common Factor or GCF).
• Techniques 2 and 3: The adding and multiplying method (finding two numbers that add to b and multiply to c).
• Technique 4: Difference of squares. If both terms are perfect squares, then you may find their square root and write it out as ((square root of term 1) - (square root of term 2)) × ((square root of term 1) + (square root of term 2)).
• Technique 5: a² ± 2ab + b² turns into (a ± b)².

Behavior of Discriminants

The discriminant is the b² - 4ac part of the quadratic formula.

• If the discriminant is positive, there are two different real roots.
• If the discriminant is zero, there is one root.
• If the discriminant is negative, there are no real roots.

Finding the Vertex in Standard Form

Method 1: Factoring and Zeroes

1. Factor to get the equation into factored form.
2. Now that you have the zeroes, add them up and divide the sum by 2; this will give you the axis (the x-coordinate of the vertex).
3. Substitute the axis into the equation to get the y-coordinate.

Method 2: Completing the Square

1. Factor the number that is in front of the x² out of the first two terms.
2. Determine the perfect square number by calculating (b/2)².
3. Add and subtract the number inside the brackets.
4. Factor the first three numbers in the bracket.
5. Add up the leftovers (multiply the subtracted perfect square which we found earlier by the a value which we factored out at first, then add this new number to the number which was excluded from the first factoring).