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Algebra 1 Unit 7: Quadratic functions & equations

Quadratic functions make U-shaped graphs called parabolas. Solving a quadratic equation finds where the parabola crosses the x-axis.

Covers: Graphs of parabolas, vertex and axis of symmetry, solving by factoring, square roots and the quadratic formula, projectile problems

Study guide

A quadratic function y = ax² + bx + c graphs as a parabola. It opens up when a > 0 (with a minimum) and down when a < 0 (with a maximum). The vertex is the turning point, and the axis of symmetry is the vertical line through it, x = −b/(2a). Plug that x back in to get the vertex's y-value. The y-intercept is c, and the x-intercepts (zeros, roots) are where y = 0.

There are several ways to solve ax² + bx + c = 0. Factoring uses the zero product property: if (x − 2)(x + 5) = 0, then x = 2 or x = −5. Square roots work when there's no x term: x² = 25 gives x = ±5. The quadratic formula, x = [−b ± √(b² − 4ac)]/(2a), always works. The discriminant b² − 4ac tells how many real solutions there are: positive means two, zero means one, negative means none.

Projectile problems use a model like h(t) = −16t² + v₀t + h₀ (feet) or −4.9t² + v₀t + h₀ (meters). The vertex gives the maximum height and when it happens. Setting h(t) = 0 and taking the positive solution gives when the object hits the ground.

To complete the square for x² + bx, add (b/2)² to make a perfect square: x² + bx + (b/2)² = (x + b/2)². When you solve an equation this way, add the same amount to both sides, then take square roots and include ±. Vertex form y = a(x − h)² + k shows the vertex (h, k) directly. Completing the square turns standard form into vertex form, so equations written in either form are equivalent.

When the discriminant isn't a perfect square, leave the answer in simplest radical form: simplify the square root and then reduce the whole fraction, as in (4 ± 2√3)/2 = 2 ± √3.

Key ideas

• y = ax² + bx + c opens up if a > 0, down if a < 0.
• Axis of symmetry: x = −b/(2a); the vertex lies on it.
• Solve by factoring, taking square roots, completing the square or the quadratic formula.
• x² = k has two solutions ±√k when k > 0.
• In projectile problems, h = 0 means the object is on the ground.

Formulas and rules

Common mistakes

• Forgetting the ± when taking a square root.
• Not setting the equation equal to 0 before factoring.

Words to know

Axis of symmetry of y = ax² + bx + c

x = −b/(2a)

Quadratic formula

x = [−b ± √(b² − 4ac)]/(2a)

Zero product property

If ab = 0, then a = 0 or b = 0.

Square root property

x² = k gives x = ±√k.

Leading coefficient a < 0 (parabola)

The parabola opens downward.

Parabola

The U-shaped graph of a quadratic function.

Vertex

The turning point of a parabola: its max or min.

Zeros (roots)

x-values where a function equals 0: the x-intercepts.

Discriminant

b² − 4ac: positive → 2 solutions, 0 → 1, negative → none real.

Completing the square

Add (b/2)² to make a perfect-square trinomial.

Projectile model

h(t) = −16t² + v₀t + h₀ (feet); the vertex gives the max height.

Sample practice questions

37 more questions like these, from level 1 to 5, are in the practice set.