Definition. A parabola is the set of points equidistant from a given point $P$ and a given line $L$ (not containing the given point).


Definition. A quadratic function is a function of the form $f(x)=ax^2+bx+c$.

Much of the work we do with quadratics consists of making connections between geometric and algebraic features. When we change the picture, what happens to the formula? When we change the formula, what happens to the picture?

Resources: Feel free to use an online calculator, like Desmos or GeoGebra to help test hypotheses and explore different formulas and graphs.

Question A: What is the relationship between each of the basic geometric transformations (translation, reflection, rotation, dilation) and the formula for a parabola?

  • For starters, look at a translation. Begin with the basic graph of $y=x^2$. Choose a translation (like “shift up 5 and shift right 2”). What is the formula for the parabola obtained by applying the translation?

Question B: What is the connection between the geometric definition of parabola (focus, directrix) and the formula for the parabola?

  • For starters, choose a point P on the y-axis to be the focus, and choose a horizontal line to be the directrix. Sketch the parabola. Can you find the equation?

Question C: What is the connection between the roots of a quadratic function and the formula for the function?

  • Make up a quadratic function. Find the roots. Can you re-write the formula for the quadratic function using only the roots as constants?

Question D: How do the following algebraic processes relate to the questions above?

  • Factoring
  • Completing the square