MYP Standard Mathematics 5.3 Quadratic Functions Notes

1. In the previous article, we covered quadratic expressions.
2. Now, we can talk about quadratic functions.
3. A "U-shaped" (or upside-down U-shaped) relationships such as projectiles, arches, and optimization problems.

Key Features Of Any Quadratic Graph

When you sketch or analyze a quadratic, you nearly always focus on three main characteristics:three main characteristics:

1. Intercepts (where the graph crosses the axes)
2. Vertex (the turning point)
3. Axis of symmetry (the vertical line through the vertex)

How The Parameters $a$, $b$, And $c$ Affect The Parabola

1. In standard form, $$y=ax^2+bx+c,\quad a\neq 0$$
2. $a$ controls opening and steepness:
   1. If $a>0$, the parabola opens upward (minimum point).
   2. If $a<0$, the parabola opens downward (maximum point).
   3. The larger $|a|$ is, the narrower (steeper) the parabola; if $0<|a|<1$, it is wider.
3. $c$ is the y-intercept (because when $x=0$, $y=c$).
4. $b$ influences where the vertex sits horizontally, and therefore the axis of symmetry.

Intercepts: Fast Ways To Find Them

Finding The y-Intercept

1. Set $x=0$:
2. So the y-intercept is $(0,c)$.

Finding x-Intercepts By Solving $y=0$

1. To find x-intercepts, set $y=0$, set $y=0$: $$ax^2+bx+c=0$$
2. How you solve depends on the form of the quadratic.
   1. If it factorizes, you can use the zero-product property.
   2. If it does not factorize nicely, you can still find intercepts using a method like the quadratic formula (often taught later), or by completing the square / technology.

Factorized Form Makes Zeros Visible

1. A quadratic written in factorized form looks like $$y=a(x-r_1)(x-r_2)$$ where $r_1$ and $r_2$ are the zeros.
2. Setting $y=0$ gives
3. So the x-intercepts are $x=r_1$ and $x=r_2$ (points $(r_1,0)$ and $(r_2,0)$).

The Vertex Lies Halfway Between The x-Intercepts

1. If a parabola has two x-intercepts $xx-intercepts $x_1$ and $x_2$, then the axis of symmetry passes exactly halfway between them.
2. That means the x-coordinate of the vertex is the average: $$x_v=\frac{x1+x_2}{2}$$
3. This is a powerful checking tool when you can see or calculate the intercepts.checking tool when you can see or calculate the intercepts.

The Vertex x-Coordinate In Standard Form Is $x

1. For any quadratic in standard form $y=ax^2+bx+c$, the vertex x-coordinate is $$x_v=-\frac{b}{2a}$$
2. This formula works whether $a=1$, $a=-1$, or any other nonzero value.

Finding The Vertex y-Coordinate Once You Know $x_v$

Once you have $x_v$, substitute it back into the function: $$y_v=f(x_v)$$

So the vertex is $(x_v, f(x_v))$.

Vertex Form Shows The Vertex Immediately

A quadratic in vertex form is $$y=a(x-h)^2+k,\quad a\neq 0$$

From this form:

1. The vertex is $(h,k)$.
2. The axis of symmetry is $x=h$.

Translation Up And Down

Adding or subtracting a constant changes $k$:

1. $y=(x-2)^2+4$ is the graph of $y=(x-2)^2$ shifted up 4 units, vertex $(2,4)$, and it has no x-intercepts.
2. $y=(x-2)^2-3$ is shifted down 3 units, vertex $(2,-3)$, and it has two x-intercepts.

When There Is Only One x-Intercept (A Repeated Root)

1. Sometimes a parabola touches the x-axis at exactly one point, at the vertex.
2. This happens when the quadratic has a repeated factor.
3. In general: $$y=a(x-h)^2$$
   1. Vertex is $(h,0)$
   2. There is exactly one unique x-intercept, $x=h$, but it is a repeated zero.

Three Equivalent Forms And When Each Is Best

Quadratics are commonly represented in three forms:

1. Standard form: $$y=ax^2+bx+c$$
   1. Best for spotting the y-intercept ($c$) and using algebraic methods like completing the square.
2. Factorized form: $$y=a(x-r_1)(x-r_2) \text{ or } y=a(x-h)^2$$
   1. Best for finding x-intercepts quickly.
3. Vertex form: $$y=a(x-h)^2+k$$
   1. Best for reading the vertex and axis of symmetry immediately, and understanding translations.