Parabola - wykres funkcji f(x)=ax^2 - Matematyka Liceum i Technikum

The Parabola Function $f(x) = ax^2$
📌 The graph of the function $f(x) = ax^2$, where $a \neq 0$, is called a parabola.
📐 Key features include the vertex always at the origin (0, 0) and the Y-axis acting as the axis of symmetry.
⬆️ If $a > 0$, the parabola opens upwards, and the range of values is $[0, +\infty)$.
⬇️ If $a < 0$, the parabola opens downwards, and the range of values is $(-\infty, 0]$.
📏 The magnitude of the coefficient $a$ determines the parabola's steepness; a larger $|a|$ results in sharper arms, while $|a| < 1$ causes the arms to spread apart.

Graph Shifting and Transformation
📐 Shifting the graph of $f(x) = ax^2$ by a vector with coordinates $(p, q)$ results in a new function $g(x)$ with the formula $g(x) = a(x - p)^2 + q$.
🔄 The vertex of the new parabola $g(x)$ is located at the point $(p, q)$, and its axis of symmetry is the line $x = p$.
➕ Shifting right corresponds to a positive $p$ in the vector, leading to $(x - p)^2$ in the formula; shifting left corresponds to a negative $p$, resulting in $(x + |p|)^2$.

Application: Finding the Function Formula After Shifting
📝 Given $f(x) = x^2$ (where $a=1$) and a shift vector $(p, q) = (-3, 2)$, the resulting function is $g(x) = 1(x - (-3))^2 + 2$, simplifying to $g(x) = (x + 3)^2 + 2$.

Application: Determining the Shift Vector
🔍 If $g(x) = x^2 - 2x - 1$ was obtained by shifting $f(x) = x^2$, by rewriting $g(x)$ in vertex form $g(x) = (x - 2)^2 - 5$, the shift vector $(p, q)$ is $(2, -5)$. (Note: The transcript used $g(x) = (x-2)^2 - 1$ in its specific example calculation for the vector $(2, -1)$).

Application: Finding Unknown Coefficient 'a'
📍 To find the formula when the vertex $(p, q) = (2, -3)$ and a point $(4, 5)$ on the graph are known:
1. Start with the form $g(x) = a(x - 2)^2 - 3$.
2. Substitute the point $(4, 5)$: $5 = a(4 - 2)^2 - 3$.
3. Solve for $a$: $8 = a(2)^2$, so $8 = 4a$, yielding $a = 2$.
4. The final formula is $g(x) = 2(x - 2)^2 - 3$.

Key Points & Insights
➡️ The shape defined by $f(x) = ax^2$ is universally called a parabola, characterized by its vertex at $(0, 0)$ and symmetry about the Y-axis.
➡️ The transformation $f(x) \to g(x) = a(x - p)^2 + q$ moves the vertex from $(0, 0)$ to $(p, q)$ and shifts the axis of symmetry to $x=p$.
➡️ Pay close attention to the signs when determining $p$ and $q$ from the shifted function's formula: $x - p$ implies a shift of $+p$ (right), and $+q$ implies an upward shift.
➡️ When finding $a$ using a point and the vertex, substitute the point's coordinates into the standard shifted equation to form a solvable linear equation for $a$.

📸 Video summarized with SummaryTube.com on Mar 04, 2026, 07:36 UTC