Quadratic Functions and Equations Notes

Review basics of algebra, including expansion, simplification, like terms, and factorization, from MAV4.
Expansion: Opening up brackets (single or double).
Factorization: Writing expressions in a compact form using brackets and common factors (opposite of expansion).

Definition: An equation containing an $x^2$ term.
Three methods to solve quadratic equations:
- Taking the square root on both sides (for some cases).
- Mid-term breaking (efficient for proficient users).
- Quadratic formula (applicable in all scenarios, provided in exams).

Applicable When: You can easily isolate the squared term and directly take the square root of both sides of the equation.
Example: Consider $x^2 = 9$ . By taking the square root, you directly get $x = \pm 3$ .
Caveat: This method is straightforward but not always applicable, especially when the equation contains a linear term (like 'bx' in $ax^2 + bx + c = 0$ ).

Process: This involves factoring the quadratic equation into two binomials.
Standard Form: Start with the quadratic equation in the form $ax^2 + bx + c = 0$ .
Find two numbers such that their product equals $a \times c$ and their sum equals $b$ .
Rewrite the middle term using these two numbers and factor by grouping.
Example: For $x^2 + 5x + 6 = 0$ , find two numbers that multiply to 6 and add to 5 (2 and 3). Rewrite as \begin{aligned}x^2 + 2x + 3x + 6 &= 0 \x(x+2) + 3(x+2) &= 0 \(x+2)(x+3) &= 0\end{aligned}Therefore, $x = -2$ or $x = -3$ .

Formula: For a quadratic equation $ax^2 + bx + c = 0$ , the solutions are given by: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ .
When to Use: Use when mid-term breaking is not feasible or easily discernible.
Given in Exams: The formula is usually provided, but knowing how to apply it is crucial.
Nature of Solutions: The discriminant ( $b^2 - 4ac$ ) determines the nature of the solutions.
- If b^2 - 4ac > 0, there are two distinct real solutions.
- If $b^2 - 4ac = 0$ , there is exactly one real solution.
- If b^2 - 4ac < 0, there are no real solutions.

Standard form: $y = ax^2 + bx + c$ .
Represents a relationship between $x$ and $y$ coordinates.
Plug and play concept: If $x$ is given, find $y$ , and vice versa.
Graphical Interpretation: Points on the curve represent solutions to the equation.

Concept: This involves substituting known values into the quadratic equation to find corresponding values.
If you know $x$ , you can find $y$ by substituting $x$ into the equation.
If you know $y$ , you can find $x$ by setting the equation equal to that $y$ value and solving for $x$ .
Graphical Significance: Finding points on the curve that satisfy given conditions.

To check if a point lies on the curve, substitute the point's coordinates $(x, y)$ into the equation.
If the equation holds true (left-hand side equals the right-hand side), the point lies on the curve.
If the equation does not hold true, the point does not lie on the curve.
If LHS = RHS, point lies on the curve; otherwise, it does not.

Shape: All quadratic functions have a parabolic shape (called a parabola).
Opening: Parabolas can open upwards (smile face) or downwards (sad face).
Turning Point/Vertex: The point where the curve changes direction.
- For upward-opening parabolas, the vertex is the minimum point.
- For downward-opening parabolas, the vertex is the maximum point.
X-intercept: Points where the graph intersects the x-axis.
Y-intercept: Point where the graph intersects the y-axis.

To find the x-intercept(s), set $y = 0$ and solve for $x$ .
To find the y-intercept, set $x = 0$ and solve for $y$ .
Logic:
- X-intercepts lie on the x-axis, where $y = 0$ .
- Y-intercepts lie on the y-axis, where $x = 0$ .

Definition: A vertical line that divides the parabola into two symmetrical halves.
Property: For every point on one side of the line, there is a corresponding point on the other side at the same distance.
Location: The line of symmetry always passes through the vertex of the parabola.

Method 1: If you know both x-intercepts ( $x1$ and $x2$ ), the line of symmetry is the average of these two $x$ values: $x = \frac{x1 + x2}{2}$ .
Method 2: Using the formula $x = -\frac{b}{2a}$ (derived from the standard form of the quadratic equation).
Key Insight: The x-coordinate of the vertex is always equal to the x-value of the line of symmetry.

Value vs. Point: It's crucial to differentiate between 'point' and 'value.' Point refers to both $x$ and $y$ coordinates, while value specifically refers to the $y$ coordinate.
Maximum Value: The $y$ coordinate of the vertex in a downward-opening parabola.
Minimum Value: The $y$ coordinate of the vertex in an upward-opening parabola.
Value is just the code word for the $Y$ axis.

$y = ax^2 + bx + c$

$y = c$

The x intercepts is $x1$ and $x2$ then equation will be like:
$y = a(x-x1)(x-x2)$
X intercepts is easy to find from this equation. But Y intercept is harder to find.
To find Y intercept plug $x = 0$

$y = a(x - h)^2 + k$

The vertexs is $(h,k)$
Given two points on an axis then find the equation or model with the help of this form then plug the knowns to find the equation.

Involves creating an equation based on given conditions or graphs.
Example conditions: Passing through specific points, having a known vertex, etc.
May involve real-life scenarios like modeling the trajectory of a cricket ball.
Solve for unknown coefficients (like 'a') by substituting known points into the equation.

These can be divided into two categories - vertical and horizontal.
Vertical transformations primarily involve movements along the y-axis, which entail shifts up and down.
Horizontal shifts or tranformations takes place along an external dimension to the original plane. This is observed or measured within the transverse plane

Also known as vertical translations, vertical shifts describe vertical movement on a graph of the function.

occurs when a certain magnitude or numeral value is added external to the existing function (i.e. plus(x) + n) where “n” refers to the units of magnitude for displacement