These are my complete notes for Quadratic Translation in Algebra II.
I color-coded my notes according to their meaning - All numbered notes (which I call rules) are red, and include examples and the basis for understanding a topic. Definitions are written in green, and other important information (such as large-scale drawings that are better visualized than explained) was written in blue. All of this information is preserved on this page, with logical flow and breaks. I use ascii line drawings sparingly - If I can convey information or a graph using an image online, I will do so.
All of the knowledge present in these notes are filtered through my personal explanations for them, the result of my attempts to understand and study them from my classes. In the unlikely event there are any egregious errors, contact me at jdlacabe@gmail.com.
Summary of Quadratic Translation (Algebra II)
I. Quadratic Translation.
Fig. 1. - The Parabola:
Vertex: The furthest point to which a parabola or other function reaches within a particular restriction.
Quadratic Expression: An equation or any function expressing a parabola on a graph.
Quantity: The total in the parenthesis at all times.
Foil/Distribute: The conversion of an expression with an outer exponent ('n') into an expression repeated 'n' times, thus multiplying the expression by itself 'n' times.
Dissection of a basic quadratic equation:
┏━ Rule 1: Quantity is Horiz. & Opposite.
┏━━┻━━┓
y = -(x - 4)² + 2 <━━ Rule 1: Y-axis, normal.
┃┗━━━━━┳━━━━━┛
┃ ┃
┃ Rule 1
┃
┃
┗━━ Rule 2 for graphing, Rule 3 for Distribution.
Rule 1. Quantity, the value in the parentheses of a quadratic equation, changes the horizontal location in the opposite manner to what is expected: y = (x-4)² will translate the parabola 4 spaces to the right rather than to the left. The y-axis change comes after the parenthesis, and is done as you would expect: a -6 moves the parabola down six spaces, while +2 brings it up two. See fig. 1. for proof of these traits of the quadratic translation.
Rule 2. None of the truths of Rule 1 change when the equation becomes negative, y = -(x - 4)² + 2 is just the positive parabola upside down.
Rule 3. For -(x - 4)², the square only applies for inside the quantity, the outer negative sticks around until the end when distribution comes (opposite/reverse everything, see Rule 1).
Rule 4. The parent function (most simplified version) of all quadratic equations is f(x) = x², given the equation has an exponent no higher than 2. Transformation means to move the function along the graphical plane in some way, such as translation vertically or horizontally (as described in Rule 1).
Rule 5. The all powerful formula: g(x) = a(x - h)² + k. This formula is known as Vertex Form. 'a' is the coefficient, the number in front of x to be distributed after simplifying. 'h' is the horizontal shift (opposite, R. 1). 'k' is the vertical shift (normal, R. 1).
Axis of Symmetry: The vertical line that divides the parabola into two equal parts. h is this axis, generally.
Rule 6. In all forms, 'a' decides whether the function/parabola will be positive or negative. 'h' is the axis of symmetry and plugging in 0 for x provides the y-intercept.
Rule 7. The Standard Form equation is f(x) = ax² + bx + c. Converting to Vertex Form (Rules 5 & 6) is h = -b / 2a, and k = f(-b / 2a), plugging in h for x.
Comparison of Standard and Vertex Form:
Standard Form = f(x) = ax² + bx + c
Vertex Form = f(x) = a(x - h)² + k
Vertex a is Standard a
Vertex h is Standard (-b / 2a)
Vertex k is Standard f(plug h for x)
Rule 8. When you are dividing zero by a number, the answer will always be zero.
Rule 9. For Axis of Symmetry, always state if as x = #, not just what the # is. Remember for tests. Vertical line.
Minimum Value: The Vertex/Domain for X when the function is at its lowest point.
Maximum Value: The Vertex/Domain for X when the function is at its highest point.
Domain: The value where x exists in which X has its limits set, all real numbers within it, see above.
Range: The limits of Y, where it exists give the Min/Max value.
Rule 10. Domain is the value of x in certain imposed or natural limits, such as Max/Min values, with certain applied factors, like Decreasing or Increasing. From a vertical perspective, going up (a positive slope) is Increasing, while going down is decreasing.
Rule 11. Decreasing Domains for wave parabolas have > greater than symbols, [ 2 > x, x > -2 ]. Increasing domains use lesser than symbols, [ -3 < x, x < 4 ], Range is below. For normal y = x² parabolas, Domain = all real #s.
Rule 12. Range is the domain for Y (Domain, Range). While for parabolas X generally doesn't have a Min/Max value, Y has a Range based on the vertex. Y uses >= of <=, as it is equal and infinitely going past the Min/Max value. For a vertex of (2, -3) with a positive a, the range is y >= 3, a parabola that just stoops below and to the right of the graph for its vertex.
Rule 13. For Intercept Form f(x) = a(x - p)(x - q), converting to Vertex Form is h = (p + q) / 2, and k is plugging in h for x in f(x), just like Rule 7. See rules 5, 6 and 16.
Rule 14. Two equations use negatives, complicating matters as you have to reverse the variable based on the positive or negative nature of the # of the particular equation. For Vertex Form, reverse h from the specific equation. In Standard Form, no reversing is necessary, the equation is positive. For Intercept Form, both p and q must be reversed.
Rule 15. In equations like -x² where you have to plug in for x, the proper way to apply the number would be -(2)², applying the square first and then the negative afterwards, in the order of operations.
Rule 16. Intercept form allows you to see the x-intercepts of the equation by simply reversing p & q. This is why it is Intercept form. See Rule 13.
Rule 17. For an x-intercept of (0, 0) in Intercept form, there is a particular way to phrase it. Instead of writing g(x) = 1(x + 6)(x + 0), you cut out q from the equation (considering zero is not relevant), and place it at the front, always meaning q = 0, like so: -x(x + 6). a = -1, p = 6, q = 0. There should be no squares in the equation, see below.
Rule 18. Intercept Form is the only Parabola Form that contains no squares (rule 4). However, when there is only x-intercept, it is possible to desimplify by taking the doubled quantities of -2(x - 3)(x - 3) and squaring it to make y = -2(x - 3). This may be confused for a Vertex Form equation with a k of 0, so always half-simplify to see. You will get the same answer either way.
Pass-through Point: The point in which a particular parabola equation h is used for substitution.
Prime Factor: A trinomial or other equation that can't be factored.
Rule 19. Finding the equation of a parabola from just a point it passes through and a form-specific Indentifer is possible through substitution. Simply to get a.
a. For Vertex Form, create the equation from the known vertex to get the base equation and substitute in the known pass-through point to get a. Ex: You have the base equation of y = a(x + 5)² + 9 and the PTP is (-7, -15). You'd plug in -15 for y and -7 for x, getting the equation -15 = a(-7 + 5)² + 9. Simplify, and you'll find a = 6 and y = -6(x + 5)² + 9.
b. For Intercept Form, mostly the same thing as Vertex Form; find the base equation through the given x-intercepts and simplify using the given PTP. Ex: Base equation is y = a(x - 9)(x - 1). Plug in PTP of (0, -18), get -18 = a(0 - 9)(0 - 1). Simplifying proves a = -2.
c. Convert Standard Form to vertex and do it. Nothing special.
Rule 20. f(x) just means y, written in function form. So is g(x). They are all the same.
Rule 21. The Square root, √, is also known as a radical. √27 = Radical Twenty Seven. For the regular square root, it takes two to get out of the square using the 28|2 method, 2√7. For cube roots, it takes three to get out, turning 3√√3 for 27 into just 3. For fractions like √16 / 25, all you have to do is square root the numerator and the denominator, getting a result of 4/5. You have to apply the method to the numerator and the denominator.
