Algebra
The Algebra domain of the Georgia Milestones Algebra EOC tests how well you create, solve, and interpret linear, quadratic, and exponential relationships. Expect items on equations and inequalities, systems, sequences, and quadratic features, often set in context so that a parameter or solution must be explained rather than just computed.
Core rules and concepts
- Linear relationships: slope is (m=\frac{y_2-y_1}{x_2-x_1}). Know slope-intercept (y=mx+b), standard (Ax+By=C), and point-slope (y-y_1=m(x-x_1)). Parallel lines share a slope; perpendicular slopes are negative reciprocals.
- Systems: substitution and elimination are the main tools. A system has one solution when the lines intersect, no solution when they are parallel, and infinitely many when the equations describe the same line.
- Inequalities: reversing the inequality symbol when multiplying or dividing by a negative number is a required step, not an optional check.
- Quadratics: for (y=ax^2+bx+c), the vertex occurs at (x=-\frac{b}{2a}). Factored form reveals zeros; the discriminant (b^2-4ac) tells how many real roots exist: positive gives two, zero gives one, negative gives none.
- Exponential and sequences: (y=a\cdot b^x) grows when (b>1) and decays when (0<b<1). Arithmetic sequences use (a_n=a_1+(n-1)d); geometric sequences use (a_n=a_1\cdot r^{,n-1}).
Worked example
A ball's height in feet is modeled by (h(t)=-16t^2+64t+5). Find its maximum height. The quadratic opens downward because (a=-16<0), so the vertex is the maximum. Use (t=-\frac{b}{2a}=-\frac{64}{2(-16)}=2). Substitute: (h(2)=-16(4)+64(2)+5=-64+128+5=69). The maximum height is 69 feet at (t=2) seconds. The reasoning matters: the vertex formula locates the time, and substitution converts that time into the requested height.
Traps that cost marks
- Dropping the negative sign when computing a perpendicular slope or when applying the quadratic formula.
- Solving (x^2=k) as only (x=\sqrt{k}) instead of (x=\pm\sqrt{k}).
- Treating an exponential model as linear because the input values increase by 1.
- Giving both roots of a quadratic in a context where one root is impossible, such as negative time or a negative length, without rejecting it.
- Forgetting that a system with identical equations has infinitely many solutions, not one.