Chapter 4: Quadratics and Polynomial Functions

Quadratics first: parabolas, zeros, vertex form, factoring, and the classic projectile model.

Problems Covered

Thrown Ball ParabolaMaximum height from vertex form. Garden Border ZerosFactoring to find intercepts. Quadratic Formula RescueUse the formula when factoring is not clean. Rational Function HolesPlanned placeholder for holes and asymptotes.

4.1 Quadratics and Parabolas

A quadratic function has degree 2. Its graph is a parabola, and the vertex marks the turning point.

f(x)=ax^2+bx+c
Core Concept

For vertex form, read the turning point directly from a(x-h)^2+k.

Worked Example 4.1.1 — Thrown Ball Parabola

A ball’s height is modeled by h(t)=-(t-3)^2+25. Find the maximum height.

  • Read the vertex from vertex form.h(t)=a(t-h)^2+k
  • Identify the vertex coordinates.(h,k)=(3,25)
  • The parabola opens downward, so the vertex is a maximum.\max h(t)=25

The maximum height is 25 units.

4.2 Zeros by Factoring

Zeros are input values that make the output equal zero. Factoring turns a quadratic equation into smaller equations.

Core Concept

To solve a factored quadratic, set each factor equal to zero.

Worked Example 4.2.1 — Garden Border Zeros

Solve x^2-7x+12=0.

  • Factor the quadratic.x^2-7x+12=(x-3)(x-4)
  • Set each factor equal to zero.x-3=0 \quad \text{or} \quad x-4=0
  • Solve the two linear equations.x=3 \quad \text{or} \quad x=4

4.3 Quadratic Formula

The quadratic formula solves any quadratic equation in standard form, including equations that do not factor nicely.

x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}
Core Concept

When factoring is not obvious, identify a, b, and c, then substitute into the quadratic formula.

Worked Example 4.3.1 — Quadratic Formula Rescue

Solve 2x^2+3x-5=0.

  • Identify the coefficients.a=2,\quad b=3,\quad c=-5
  • Substitute into the formula.x=\frac{-3\pm\sqrt{3^2-4(2)(-5)}}{2(2)}
  • Simplify the discriminant.x=\frac{-3\pm\sqrt{49}}{4}
  • Split the two solutions.x=1 \quad \text{or} \quad x=-\frac{5}{2}

4.4 Rational Functions

Rational function pages will cover holes, vertical asymptotes, horizontal asymptotes, and sign charts. The classic problems here will include removable gaps and long-run behavior.