Overview
This lecture explains how to identify, interpret, and sketch parabolas, focusing on transformations like shifting and the process to find key points such as intercepts and turning points.
Recognizing Parabolas
- A parabola is represented by the equation y = x² or similar forms where x is the base, not the exponent.
- Do not confuse parabolas (x²) with exponential graphs (x in the exponent).
Transformations of Parabolas
- Adding or subtracting outside the squared term (e.g., y = x² + 1) shifts the graph vertically.
- Adding or subtracting inside brackets with x (e.g., y = (x - 2)²) shifts the graph horizontally (opposite direction: minus means right, plus means left).
- The coefficient in front of x² (a) affects: a > 0 "smiling" (opens upward), a < 0 "sad" (opens downward).
- The greater the |a| value, the narrower the parabola; smaller |a| makes it wider.
Sketching Parabolas: Practice Examples
- Vertical shifts modify the y-coordinate of the turning point; no change in x.
- Horizontal shifts move the turning point left/right; minus in the brackets means right, plus means left.
- The sign of the coefficient determines if the graph opens up or down.
Steps to Draw a Parabola
- Find x-intercepts by setting y = 0 and solving (use factoring or the quadratic formula if needed).
- Find the y-intercept by setting x = 0.
- Find the turning point using either the midpoint of the x-intercepts or the formula x = -b/(2a).
- To find the y of the turning point, substitute its x value into the original equation.
Turning Point Form
- If the equation is written as y = a(x - h)² + k, the turning point is (h, k).
- Use this form to find the turning point quickly if available.
Key Terms & Definitions
- Parabola — a u-shaped graph of the equation y = ax² + bx + c.
- Turning Point — the vertex of the parabola, the highest or lowest point.
- x-intercept — the point(s) where the graph crosses the x-axis (y = 0).
- y-intercept — the point where the graph crosses the y-axis (x = 0).
- Vertical Shift — moving the graph up or down.
- Horizontal Shift — moving the graph left or right.
- Coefficient a — determines the direction and width of the parabola.
Action Items / Next Steps
- Practice sketching parabolas using transformations (vertical and horizontal shifts).
- Learn how to apply the quadratic formula for x-intercepts.
- Review how to convert equations into turning point form.
- Prepare for more examples in the next lesson.