Combined Translations: Quadratic Functions
Combined translations of quadratic functions apply both a horizontal shift h and a vertical shift k to produce the vertex form f(x) = (x − h)² + k, covered in Grade 11 enVision Algebra 1 (Chapter 10: Working With Functions). The process involves completing the square to convert standard form to vertex form, then reading off h and k to identify the vertex (h, k). Graphing proceeds by applying the horizontal shift first, then the vertical shift to the parent parabola f(x) = x². This approach makes graphing any quadratic systematic.
Key Concepts
How To Graph a quadratic function using translations.
Step 1. Rewrite the function in $f(x) = (x h)^2 + k$ form by completing the square. This is also known as the vertex form.
Common Questions
What is vertex form of a quadratic function?
Vertex form is f(x) = (x − h)² + k, where the vertex is at (h, k) and the parabola is a translation of the parent function f(x) = x².
How do you graph a quadratic using combined translations?
Step 1: Rewrite in vertex form by completing the square. Step 2: Identify h (horizontal shift) and k (vertical shift). Step 3: Apply horizontal shift, then vertical shift to the parent parabola.
What is the vertex of f(x) = (x − 3)² + 5?
The vertex is (3, 5). The graph is shifted 3 units right and 5 units up from the parent parabola.
How do you complete the square to find vertex form?
Factor out any leading coefficient from the x² and x terms, add and subtract (half the x coefficient)² inside, then group the perfect square trinomial and simplify.
Does the direction the parabola opens depend on the vertex form?
The basic vertex form (x − h)² + k opens upward. If the coefficient is negative — like −(x − h)² + k — it opens downward.
What is the vertex of f(x) = x² − 6x + 11?
Complete the square: f(x) = (x − 3)² + 2. The vertex is (3, 2).