🟣 Graphing Quadratic Functions
Visualize parabolas! Learn to graph quadratic functions, identify key features, and understand transformations.
📐 Understanding the Graph of a Quadratic Function
A quadratic function is any function that can be written in the form:
$$f(x)=ax^2+bx+c$$
Parabola Properties
- The graph of a quadratic function is a parabola.
- If $a>0$, the parabola opens upwards (U-shape) – has a minimum.
- If $a<0$, the parabola opens downwards (∩-shape) – has a maximum.
- The width of the parabola is determined by $|a|$: larger values make it narrower.
Example: Graph $f(x)=x^2-4x+3$. Since $a=1>0$, the parabola opens upward.
🎯 Key Features of a Parabola
1️⃣ Vertex – The Turning Point
The vertex is the highest or lowest point of the parabola.
Formula to find the x-coordinate of the vertex:
$$x=\frac{-b}{2a}$$
Then substitute $x$ back into the function to find $y$.
Example: Find the vertex of $f(x)=x^2-6x+5$
$x=\frac{6}{2}=3$
Then, $f(3)=9-18+5=-4$.
Vertex = $(3,-4)$
2️⃣ Axis of Symmetry
The vertical line through the vertex, given by the same formula $x=\frac{-b}{2a}$, divides the parabola into mirror images.
Example: For $f(x)=2x^2-4x+1$, the axis of symmetry is $x=1$.
This means the parabola is symmetric about the line $x=1$.
3️⃣ X-Intercepts (Roots)
The x-intercepts occur where $f(x)=0$. Solve the equation $ax^2+bx+c=0$ to find them.
Example: Find the x-intercepts of $f(x)=x^2-5x+6$
Factor: $(x-2)(x-3)=0$
Thus, $x=2$ and $x=3$.
X-intercepts: $(2,0)$ and $(3,0)$
4️⃣ Y-Intercept
The y-intercept is found by evaluating $f(0)$. It equals the constant term $c$.
Example: For $f(x)=3x^2-5x+2$, $f(0)=2$.
Y-intercept: $(0,2)$
5️⃣ Vertex Form vs. Standard Form
We often switch between forms to make graphing easier:
- Standard Form: $f(x) = ax^2 + bx + c$ (Good for finding y-intercept)
- Vertex Form: $f(x) = a(x-h)^2 + k$ (Good for finding vertex $(h,k)$)
Example: $f(x) = 2(x-1)^2 - 3$
Vertex is $(1, -3)$. Axis of symmetry is $x=1$. Since $a=2>0$, it opens upward.
📝 How to Graph a Quadratic Function
Follow these steps to graph any quadratic function:
Example: Step-by-Step
Graph $f(x)=x^2-4x+3$
- Find the vertex:
$x=\frac{-(-4)}{2(1)}=\frac{4}{2}=2$ and $f(2)=4-8+3=-1$
Vertex = $(2,-1)$
- Find the x-intercepts:
Factor: $(x-1)(x-3)=0$
Thus, $x=1$ and $x=3$
X-intercepts: $(1,0)$ and $(3,0)$
- Find the y-intercept:
$f(0)=3$
Y-intercept: $(0,3)$
- Plot the points and draw a smooth parabola through them.
🔄 Transformations of Quadratic Functions
Understanding transformations helps you quickly sketch parabolas!
From Parent Function $y=x^2$
- Vertical shift: $y=x^2+k$ shifts up ($k>0$) or down ($k<0$)
- Horizontal shift: $y=(x-h)^2$ shifts right ($h>0$) or left ($h<0$)
- Vertical stretch/compression: $y=ax^2$ stretches if $|a|>1$, compresses if $0<|a|<1$< /li>
- Reflection: $y=-x^2$ reflects across the x-axis
Example: $y=-2(x+3)^2+5$
This is the parent function $y=x^2$:
- Shifted left 3 units
- Shifted up 5 units
- Stretched vertically by factor of 2
- Reflected across x-axis
Vertex: $(-3,5)$ and opens downward
🎯 Practice Questions
Master graphing techniques!
🔥 Challenge Questions
Push your graphing skills!