📚 Understanding Vertical and Horizontal Shifts in Vertex Form
The vertex form of a quadratic function is expressed as $f(x) = a(x - h)^2 + k$, where $(h, k)$ represents the vertex of the parabola. The values of $h$ and $k$ dictate the horizontal and vertical shifts, respectively, relative to the basic parabola $f(x) = x^2$. Let's break down each shift:
- ➡️ Horizontal Shifts: The value of $h$ within the vertex form $f(x) = a(x - h)^2 + k$ controls the horizontal shift. It's crucial to note that the shift is opposite the sign of $h$. For instance:
- 🔍 If $h > 0$, the parabola shifts to the right by $h$ units. Example: $f(x) = (x - 3)^2$ shifts the graph of $f(x) = x^2$ three units to the right.
- ⬅️ If $h < 0$, the parabola shifts to the left by $|h|$ units. Example: $f(x) = (x + 2)^2$ shifts the graph of $f(x) = x^2$ two units to the left.
- ⬆️ Vertical Shifts: The value of $k$ in the vertex form $f(x) = a(x - h)^2 + k$ determines the vertical shift. The shift corresponds directly with the sign of $k$:
- 📈 If $k > 0$, the parabola shifts upward by $k$ units. Example: $f(x) = x^2 + 4$ shifts the graph of $f(x) = x^2$ four units upward.
- 📉 If $k < 0$, the parabola shifts downward by $|k|$ units. Example: $f(x) = x^2 - 5$ shifts the graph of $f(x) = x^2$ five units downward.
- 💡Combined Shifts: When both $h$ and $k$ are present, the parabola undergoes both horizontal and vertical shifts. For example, $f(x) = (x - 1)^2 + 3$ shifts the graph of $f(x) = x^2$ one unit to the right and three units upward, placing the vertex at $(1, 3)$.
🧮 Examples in Vertex Form
Let's examine a few examples to solidify your understanding:
- Example 1: $f(x) = (x + 2)^2 - 1$. Here, $h = -2$ and $k = -1$. This indicates a shift of 2 units to the left and 1 unit downward, placing the vertex at $(-2, -1)$.
- Example 2: $f(x) = (x - 3)^2 + 4$. Here, $h = 3$ and $k = 4$. This indicates a shift of 3 units to the right and 4 units upward, placing the vertex at $(3, 4)$.
- Example 3: $f(x) = x^2 - 5$. Here, $h = 0$ and $k = -5$. This indicates no horizontal shift and a shift of 5 units downward, placing the vertex at $(0, -5)$.
📊 Table Summarizing Shifts
To further clarify, consider the following table:
| Function |
Horizontal Shift |
Vertical Shift |
Vertex |
| $f(x) = (x - 2)^2 + 3$ |
2 units right |
3 units up |
$(2, 3)$ |
| $f(x) = (x + 1)^2 - 4$ |
1 unit left |
4 units down |
$(-1, -4)$ |
| $f(x) = (x - 5)^2$ |
5 units right |
No shift |
$(5, 0)$ |
| $f(x) = x^2 + 2$ |
No shift |
2 units up |
$(0, 2)$ |
✍️ Practice Quiz
- 📝 Identify the vertex of the quadratic function: $f(x) = (x - 4)^2 + 1$.
- 🧮 Describe the shifts applied to $f(x) = x^2$ to obtain $f(x) = (x + 3)^2 - 2$.
- 📈 Write the equation of a quadratic function with a vertex at $(1, -5)$.
- 🤔 Determine the horizontal and vertical shifts for the function $f(x) = (x - 7)^2 + 6$.
- 💡 If a quadratic function has a vertex at $(-2, 3)$, what is its vertex form equation?