Linear Equations & Lines
A linear relationship changes at a constant rate — its graph is a straight line. Mastering lines is the foundation for everything from SAT word problems to the slope fields of calculus.
A linear equation in one variable has the form \( ax+b=0 \) with \( a\neq 0 \); its single solution is \( x=-\dfrac{b}{a} \). In two variables, \( y=mx+b \) describes a line of slope \( m \) and \( y \)-intercept \( b \).
The slope measures steepness as rise over run, \( m=\dfrac{\Delta y}{\Delta x}=\dfrac{y_2-y_1}{x_2-x_1} \). Drag the sliders to see how \( m \) tilts the line and \( b \) slides it up or down — the amber dots mark where it crosses the axes.
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Three forms of a line
Solve \( 3x+5=20 \). Subtract 5, then divide by 3:
Find the line through \( (1,2) \) and \( (5,10) \). Slope first, then point–slope with \( (1,2) \):
Quadratic Functions
Quadratics introduce curvature. Their U-shaped graphs — parabolas — appear in projectile motion, optimization, and a large share of every standardized test.
A quadratic function has the form \( f(x)=ax^2+bx+c \) with \( a\neq 0 \). Its graph is a parabola opening upward when \( a>0 \) and downward when \( a<0 \), with axis of symmetry \( x=-\dfrac{b}{2a} \).
The solutions of \( ax^2+bx+c=0 \) are The discriminant \( \Delta=b^2-4ac \) reveals the roots: \( \Delta>0 \) gives two real roots, \( \Delta=0 \) one repeated root, and \( \Delta<0 \) no real roots.
In the explorer, \( a \), \( b \), and \( c \) reshape the parabola live. The amber vertex sits on the dashed axis of symmetry, and when the discriminant is non-negative the amber roots appear exactly where the curve meets the \( x \)-axis.
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Three forms of a quadratic
Solve \( x^2-5x+6=0 \). Find two numbers multiplying to \( 6 \) and adding to \( -5 \): namely \( -2 \) and \( -3 \).
Solve \( 2x^2+3x-2=0 \). Here \( a=2,\ b=3,\ c=-2 \), so \( \Delta=9+16=25 \):
Exponents & Polynomials
Exponent rules and polynomial structure are the algebraic “grammar” that keeps longer manipulations clean and error-free.
\( a^m a^n=a^{m+n} \), \(\quad \dfrac{a^m}{a^n}=a^{m-n} \), \(\quad (a^m)^n=a^{mn} \), \( a^0=1\ (a\neq 0) \), \(\quad a^{-n}=\dfrac{1}{a^n} \), \(\quad a^{1/n}=\sqrt[n]{a}. \)
Special products & factoring
Simplify \( \dfrac{x^5 \cdot x^2}{x^3} \). Add exponents on top, then subtract:
Factor \( x^2-9 \). A difference of squares with \( a=x,\ b=3 \):
Functions & Transformations
A function is a rule that assigns exactly one output to each input. Once you know a parent graph, simple changes to its formula shift, stretch, and flip it — no re-plotting required.
A function \( f \) pairs each input \( x \) with one output \( f(x) \). The domain is the set of allowed inputs; the range is the set of resulting outputs.
Starting from a parent function \( f \), the graph of is stretched vertically by \( |a| \) (and reflected if \( a<0 \)), shifted right by \( h \), and shifted up by \( k \).
Pick a parent function and move the sliders — the faint grey curve is the original, the indigo curve is the transformed graph, and the amber dot tracks its key point at \( (h,k) \).
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How is \( y=(x-3)^2+2 \) related to \( y=x^2 \)? Matching \( a=1,\ h=3,\ k=2 \): the parabola shifts right \( 3 \) and up \( 2 \), placing the vertex at \( (3,2) \).
Find the domain of \( f(x)=\sqrt{x-1} \). The radicand must be non-negative: \( x-1\geq 0 \), so the domain is \( [1,\infty) \).
A function assigns exactly one output to each input. Transformations shift, reflect, stretch, or compress the graph of a parent function.
Describe the transformation from \( y=x^2 \) to \( y=-(x-3)^2+4 \). Right 3, reflect over \( x \)-axis, up 4. Vertex at \( (3,4) \), opens downward.
Exponential & Logarithmic Functions
Exponential functions model growth and decay — populations, interest, radioactivity. Logarithms are their inverses, and the tool that solves for an unknown exponent.
An exponential function has the form \( f(x)=a\,b^{x} \) with \( b>0,\ b\neq 1 \). It models growth when \( b>1 \) and decay when \( 0<b<1 \), with horizontal asymptote \( y=0 \) and \( y \)-intercept \( a \).
\( y=\log_b x \iff b^{y}=x \). The laws follow directly:
Move the sliders to see growth flip to decay as the base crosses \( 1 \). Tick the box to mirror the curve over \( y=x \) and reveal its inverse logarithm.
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Solve \( 2^{x}=32 \). Write \( 32 \) as a power of \( 2 \): \( 32=2^{5} \). Equal bases force equal exponents:
Evaluate \( \log_2 8 \) and expand \( \log\!\big(xy^2\big) \). Since \( 8=2^{3} \), \( \log_2 8=3 \). By the log laws,
Systems & Inequalities
A system of equations asks where two conditions hold at once. Graphically, the solution is the point where the lines cross — and sometimes there is no such point.
A solution satisfies every equation simultaneously. Two lines either cross once (one solution), run parallel (no solution), or coincide (infinitely many).
Solve by substitution (isolate one variable and plug in) or elimination (add multiples of the equations to cancel a variable). Drag the four sliders: the amber dot is the live intersection, and parallel slopes report “no solution”.
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Solve \( \begin{cases} 2x+y=7 \\ x-y=2 \end{cases} \). Add the equations to cancel \( y \): \( 3x=9 \Rightarrow x=3 \). Back-substitute: \( 3-y=2 \Rightarrow y=1 \).
Solve \( \begin{cases} y=2x+1 \\ y=2x-3 \end{cases} \). Equal slopes but different intercepts: the lines are parallel and never meet — no solution.
Solve systems by substitution, elimination, or matrices. Graph inequalities with dashed or solid boundary lines and shade the solution region.
Solve: \( x+y=5 \), \( 2x-y=4 \). Add: \( 3x=9 \Rightarrow x=3,\;y=2 \).
Sequences & Series
A sequence is an ordered list of numbers following a rule; a series is their sum. Two patterns dominate: adding a constant (arithmetic) and multiplying by a constant (geometric).
Each term adds a common difference \( d \): \( a_n=a_1+(n-1)d \). The sum of the first \( n \) terms is
Each term multiplies by a common ratio \( r \): \( a_n=a_1\,r^{\,n-1} \). The first \( n \) terms sum to \( S_n=a_1\dfrac{1-r^{n}}{1-r} \), and when \( |r|<1 \) the infinite sum converges:
Find the sum of the first \( 20 \) terms of \( 3,7,11,\dots \) Here \( a_1=3,\ d=4 \), so \( a_{20}=3+19(4)=79 \). Then
Evaluate \( 8+4+2+1+\cdots \) With \( a_1=8 \) and \( r=\tfrac12 \) (and \( |r|<1 \)):
Factoring Polynomials
Factoring rewrites a polynomial as a product of simpler factors — the key step in solving many equations.
Always take out the greatest common factor first. Difference of squares: \( a^{2}-b^{2}=(a-b)(a+b) \). Trinomial: \( x^{2}+(p+q)x+pq=(x+p)(x+q) \).
Factor \( x^{2}-5x+6 \).
Radicals & Exponents
Roots can be written as radicals or as fractional exponents; both obey the laws of exponents.
\( \sqrt{ab}=\sqrt{a}\,\sqrt{b} \), \( a^{1/n}=\sqrt[n]{a} \), and \( a^{m/n}=\sqrt[n]{a^{m}} \). Rationalise a denominator by multiplying by the conjugate.
Simplify \( \sqrt{72} \).
Rational Expressions
A rational expression is a ratio of polynomials; simplify by factoring and cancelling common factors.
Multiply \( \tfrac{a}{b}\cdot\tfrac{c}{d}=\tfrac{ac}{bd} \); divide by multiplying by the reciprocal; add over a common denominator. Exclude any value making a denominator zero.
Simplify \( \dfrac{x^{2}-9}{x-3} \).
Statistics & Data Analysis
Summarise data with measures of centre and spread, and read trends from tables and plots.
The mean is the average, the median the middle value, the mode the most frequent, and the range the spread. A line of best fit models a linear trend in a scatter plot.
Find the mean of \( 3,\,6,\,9,\,12 \).
Practice set
Fifty mixed problems spanning all seven chapters, ordered roughly from linear equations to series and tagged by difficulty. Click any problem to reveal a full worked solution.
Solve \( 4x-7=2x+9 \).
BasicCollect like terms: \( 4x-2x=9+7 \), so \( 2x=16 \).
\[ x=8. \]Find the slope of the line through \( (1,2) \) and \( (5,10) \).
BasicSolve \( x^2-7x+12=0 \).
BasicFactor: two numbers with product \( 12 \), sum \( -7 \) are \( -3,-4 \).
\[ (x-3)(x-4)=0 \Rightarrow x=3 \ \text{or}\ 4. \]Solve \( 2x^2+5x-3=0 \) using the quadratic formula.
Intermediate\( a=2,\ b=5,\ c=-3 \), so \( \Delta=25+24=49 \):
\[ x=\frac{-5\pm 7}{4}=\tfrac12 \ \text{or}\ -3. \]Simplify \( \dfrac{(x^3y^2)(x^2y^4)}{x\,y^3} \).
IntermediateAdd exponents in the numerator, then subtract the denominator's:
\[ x^{3+2-1}\,y^{2+4-3}=x^4 y^3. \]Factor \( 9x^2-25 \).
IntermediateDifference of squares with \( a=3x,\ b=5 \):
\[ 9x^2-25=(3x-5)(3x+5). \]Find the vertex and axis of symmetry of \( y=x^2-6x+5 \).
AdvancedAxis: \( x=-\dfrac{b}{2a}=\dfrac{6}{2}=3 \). Then \( y=9-18+5=-4 \).
\[ \text{Vertex } (3,-4), \quad \text{axis } x=3. \]Describe the transformation from \( y=x^2 \) to \( y=(x-2)^2-5 \).
BasicWith \( h=2,\ k=-5 \): shift right \( 2 \) and down \( 5 \).
\[ \text{Vertex moves to } (2,-5). \]Solve \( 3^{x}=81 \).
BasicWrite \( 81=3^{4} \); equal bases give equal exponents:
\[ 3^{x}=3^{4}\Rightarrow x=4. \]Solve \( \begin{cases} x+y=10 \\ x-y=4 \end{cases} \).
IntermediateAdd the equations: \( 2x=14\Rightarrow x=7 \). Then \( 7+y=10\Rightarrow y=3 \).
\[ (x,y)=(7,3). \]Find the 6th term and the infinite sum of \( 27,9,3,\dots \)
IntermediateGeometric with \( a_1=27,\ r=\tfrac13 \). The 6th term is \( a_6=27\big(\tfrac13\big)^{5}=\tfrac{27}{243}=\tfrac19 \). Since \( |r|<1 \),
\[ S_\infty=\frac{27}{1-\tfrac13}=\frac{27}{2/3}=40.5. \]Solve \( 5x-3=2x+12 \).
Basic\( 3x=15 \):
\[ x=5. \]Solve \( 2(x-4)=10 \).
BasicDivide by 2: \( x-4=5 \).
\[ x=9. \]Find the slope of the line through \( (2,3) \) and \( (6,11) \).
BasicWrite the equation of the line through \( (0,-2) \) with slope \( 3 \).
IntermediateSlope–intercept with \( b=-2 \):
\[ y=3x-2. \]Find the \( x \)-intercept of \( 2x-5y=10 \).
IntermediateSet \( y=0 \): \( 2x=10 \).
\[ (5,0). \]Are \( y=2x+1 \) and \( y=2x-4 \) parallel, perpendicular, or neither?
IntermediateEqual slopes \( (2=2) \), different intercepts:
\[ \text{parallel}. \]What is the slope of any line perpendicular to \( y=\tfrac{2}{3}x+1 \)?
IntermediateNegative reciprocal of \( \tfrac23 \):
\[ -\frac{3}{2}. \]Solve \( x^2=49 \).
BasicSolve \( x^2-9x+20=0 \).
BasicFactor: \( (x-4)(x-5)=0 \).
\[ x=4 \ \text{or}\ 5. \]Solve \( x^2+6x+9=0 \).
IntermediatePerfect square \( (x+3)^2=0 \):
\[ x=-3 \ \text{(double root)}. \]How many real roots does \( 2x^2-4x+5=0 \) have?
IntermediateDiscriminant \( \Delta=16-40=-24<0 \):
\[ \text{no real roots}. \]Solve \( 3x^2-12=0 \).
Basic\( x^2=4 \):
\[ x=\pm 2. \]Find the vertex of \( y=x^2+4x+1 \).
Intermediate\( x=-\tfrac{b}{2a}=-2 \), then \( y=4-8+1=-3 \):
\[ (-2,-3). \]Solve \( x^2-4x-1=0 \).
AdvancedQuadratic formula, \( \Delta=16+4=20 \):
\[ x=\frac{4\pm\sqrt{20}}{2}=2\pm\sqrt5. \]Complete the square: \( x^2+6x+5 \).
AdvancedHalf of 6 is 3; add and subtract \( 9 \):
\[ x^2+6x+5=(x+3)^2-4. \]Simplify \( x^4\cdot x^3 \).
BasicSimplify \( \dfrac{x^5}{x^2} \).
BasicSimplify \( \big(x^2\big)^3 \).
BasicEvaluate \( 2^{-3} \).
BasicExpand \( (x+4)^2 \).
IntermediateFactor \( x^2-16 \).
IntermediateDifference of squares:
\[ (x-4)(x+4). \]Factor \( x^2+7x+12 \).
IntermediateTwo numbers multiplying to 12, adding to 7 are 3 and 4:
\[ (x+3)(x+4). \]If \( f(x)=2x^2-1 \), find \( f(3) \).
BasicFind the domain of \( f(x)=\sqrt{x-3} \).
IntermediateNeed \( x-3\ge 0 \):
\[ [3,\infty). \]Find the domain of \( f(x)=\dfrac{1}{x-2} \).
IntermediateThe denominator cannot be zero:
\[ x\neq 2. \]Describe the transformation from \( y=|x| \) to \( y=|x+2|-3 \).
Intermediate\( h=-2,\ k=-3 \):
\[ \text{left 2, down 3}. \]Is \( f(x)=x^3 \) even, odd, or neither?
Intermediate\( f(-x)=-x^3=-f(x) \):
\[ \text{odd}. \]Solve \( 5^{x}=125 \).
Basic\( 125=5^3 \):
\[ x=3. \]Evaluate \( \log_3 81 \).
Intermediate\( 81=3^4 \):
\[ \log_3 81=4. \]Evaluate \( \log_2\!\big(\tfrac18\big) \).
Intermediate\( \tfrac18=2^{-3} \):
\[ \log_2\!\big(\tfrac18\big)=-3. \]Expand \( \log\!\big(x^3 y\big) \).
IntermediateSolve \( 2^{\,x+1}=16 \).
Advanced\( 16=2^4 \), so \( x+1=4 \):
\[ x=3. \]Solve \( \begin{cases} x+y=5 \\ x-y=1\end{cases} \).
IntermediateAdd: \( 2x=6\Rightarrow x=3 \), then \( y=2 \):
\[ (3,2). \]Solve \( \begin{cases} y=2x \\ x+y=9\end{cases} \).
IntermediateSubstitute: \( x+2x=9\Rightarrow x=3 \), so \( y=6 \):
\[ (3,6). \]Solve the inequality \( 3x-4>11 \).
Basic\( 3x>15 \):
\[ x>5. \]Solve \( -2x+1\le 7 \).
Intermediate\( -2x\le 6 \); dividing by \( -2 \) flips the sign:
\[ x\ge -3. \]Find the 10th term of the arithmetic sequence \( 2,5,8,\dots \)
Basic\( a_1=2,\ d=3 \):
\[ a_{10}=2+9(3)=29. \]Find the sum of the first 20 terms of \( 2,5,8,\dots \)
Intermediate\( a_{20}=2+19(3)=59 \), then \( S_{20}=\tfrac{20}{2}(2+59) \):
\[ S_{20}=10\cdot 61=610. \]Evaluate the infinite sum \( 16+8+4+\cdots \)
IntermediateGeometric with \( a=16,\ r=\tfrac12 \):
\[ S_\infty=\frac{16}{1-\tfrac12}=32. \]Test generator
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