Question

What key features of a quadratic graph can be identified and how are the graphs affected when constants or coefficients are added to the parent quadratic equations? Compare the translations to the graph of linear function.

Answer 1

Key features of a quadratic graph include the vertex, axis of symmetry, direction of opening, and intercepts.

When constants or coefficients are added to the parent quadratic equation, the graph undergoes translations.

• - Adding a constant term (e.g., "+c") shifts the graph vertically by c units, without affecting the shape or direction of the parabola.
• - Multiplying the entire equation by a constant (e.g., "a(x-h)^2") affects the steepness or stretch of the parabola. If |a| > 1, the parabola becomes narrower, while if |a| < 1, the parabola becomes wider. The sign of "a" determines whether the parabola opens upward (a > 0) or downward (a < 0).
• - Adding a linear term (e.g., "+bx") introduces a slant or tilt to the parabola, causing it to become a "quadratic equation of the second degree" or a "quadratic expression." This term affects the axis of symmetry and the vertex.

In comparison to a linear function, quadratic graphs have a curved shape and are symmetric about their axis. Linear graphs, on the other hand, are straight lines and do not have a vertex or axis of symmetry.