SAT 代数与高等数学练习题

Expressions can show fixed values, rates, grouped quantities, and operations described in words. Rewriting an expression can reveal a useful contextual meaning.

Absolute value equations can be interpreted as distance from zero or from a center point. Equations of the form $|x-a|=b$ usually have two solutions when $b>0$.

Context problems often require defining a variable, writing a one-variable linear equation, and interpreting the solution in the original situation.

Linear equations in one variable ask for the value of a single unknown that makes an equation true. Common moves include distributing, combining like terms, clearing fractions, and using inverse operations.

Linear equations in one variable ask for the value of a single unknown that makes an equation true. Common moves include distributing, combining like terms, clearing fractions, and using inverse operations.

Linear equations in one variable ask for the value of a single unknown that makes an equation true. Common moves include distributing, combining like terms, clearing fractions, and using inverse operations.

Linear equations in one variable ask for the value of a single unknown that makes an equation true. Common moves include distributing, combining like terms, clearing fractions, and using inverse operations.

Parameter questions in one-variable linear equations often ask for a value that makes an equation have one solution, no solution, or infinitely many solutions.

A linear equation in two variables can be written in forms such as $ax+by=c$ or $y=mx+b$. A solution is an ordered pair $(x,y)$ that makes the equation true.

A linear equation in two variables can be written in forms such as $ax+by=c$ or $y=mx+b$. A solution is an ordered pair $(x,y)$ that makes the equation true.

A linear equation in two variables can be written in forms such as $ax+by=c$ or $y=mx+b$. A solution is an ordered pair $(x,y)$ that makes the equation true.

A linear equation in two variables can be written in forms such as $ax+by=c$ or $y=mx+b$. A solution is an ordered pair $(x,y)$ that makes the equation true.

Parameter questions for linear equations in two variables often use a given point, intercept, or slope condition to determine a missing coefficient.

Two-variable linear equations can be interpreted through intercepts, points on a graph, and context equations with two quantities.

Function notation describes input-output relationships. Linear function questions often ask for a function value, an input that gives a specified output, a rate of change, or a parameter in a function rule.

Linear function questions often ask for a rate of change, an initial value, a missing table value, or an equation that matches a real-world relationship.

A linear function has a constant rate of change and can often be written as $f(x)=mx+b$, where $m$ is the rate of change and $b$ is the value of the function when $x=0$.

A linear function has a constant rate of change and can often be written as $f(x)=mx+b$, where $m$ is the rate of change and $b$ is the value of the function when $x=0$.

A linear function has a constant rate of change and can often be written as $f(x)=mx+b$, where $m$ is the rate of change and $b$ is the value of the function when $x=0$.

A linear function has a constant rate of change and can often be written as $f(x)=mx+b$, where $m$ is the rate of change and $b$ is the value of the function when $x=0$.

Parameter questions for linear functions may ask for an unknown slope, intercept, or coefficient from function values or rate-of-change conditions.

Tables, graphs, verbal descriptions, and equations can describe the same linear relationship. Convert between them by identifying slope and initial value.

In a linear model, the constant term is the starting value and the coefficient of the input is the rate of change. The sign of the coefficient shows increase or decrease.

Contextual algebra problems often combine units, rates, initial values, and restrictions such as budgets or nonnegative amounts.

Absolute value inequalities describe distances. Inequalities of the form $|x-a|<b$ describe values within $b$ units of $a$, while $|x-a|>b$ describes values more than $b$ units away.

Compound inequalities require the same operation on each part. Endpoint type matters when counting integers.

Context problems often translate into equations or inequalities by tracking units. Totals, budgets, minimums, and maximums determine the operation and inequality direction.

Words such as at most, at least, minimum, maximum, and budget indicate inequality direction. Integer contexts may require rounding up or down.

Inequality questions are error-prone because strict versus inclusive endpoints, negative division, compound inequalities, and integer boundaries all affect the final answer.

When solving inequalities, multiplying or dividing by a negative number reverses the inequality sign. This is a common source of wrong answers.

Linear inequalities use symbols such as $<$, $>$, $\le$, and $\ge$ to describe ranges of values. Solving them is similar to solving linear equations, except the inequality symbol reverses when both sides are multiplied or divided by a negative number.

Linear inequalities use symbols such as $<$, $>$, $\le$, and $\ge$ to describe ranges of values. Solving them is similar to solving linear equations, except the inequality symbol reverses when both sides are multiplied or divided by a negative number.

Linear inequalities use symbols such as $<$, $>$, $\le$, and $\ge$ to describe ranges of values. Solving them is similar to solving linear equations, except the inequality symbol reverses when both sides are multiplied or divided by a negative number.

Linear inequalities use symbols such as $<$, $>$, $\le$, and $\ge$ to describe ranges of values. Solving them is similar to solving linear equations, except the inequality symbol reverses when both sides are multiplied or divided by a negative number.

Parameter questions for inequalities often require identifying the coefficient that produces a stated solution set.

Linear inequalities use symbols such as $<$, $>$, $\le$, and $\ge$ to describe ranges of values. Supplemental practice should emphasize integer boundary questions, compound inequalities, strict versus inclusive endpoints, context constraints, and parameter values that determine a solution set.

Inequality questions use symbols to represent limits, minimums, maximums, and ranges. Contexts often require translating phrases such as at least, at most, fewer than, or no more than.

Linear systems with parameters often test whether two lines intersect once, never intersect, or represent the same line. Compare slopes and constants carefully.

Parameters in linear systems usually affect slopes, intercepts, or constants. Compare coefficient ratios to decide whether lines intersect once, never, or infinitely many times.

A system of linear equations can model two simultaneous conditions. Graphically, the solution is the intersection point of the two lines.

A system of two linear equations in two variables asks for values of both variables that make both equations true. Common methods include substitution, elimination, comparing equivalent equations, and interpreting the graphs as lines.

A system of two linear equations in two variables asks for values of both variables that make both equations true. Common methods include substitution, elimination, comparing equivalent equations, and interpreting the graphs as lines.

A system of two linear equations in two variables asks for values of both variables that make both equations true. Common methods include substitution, elimination, comparing equivalent equations, and interpreting the graphs as lines.

A system of two linear equations in two variables asks for values of both variables that make both equations true. Common methods include substitution, elimination, comparing equivalent equations, and interpreting the graphs as lines.

Parameter questions for systems of linear equations often involve one solution, no solution, infinitely many solutions, or a specified coordinate in the solution.

Absolute value can model distance on a number line. This viewpoint makes solution counts and tolerance statements easier to interpret.

An absolute value equation with a positive right side usually splits into two linear equations. A zero right side gives one solution, and a negative right side gives no solution.

Absolute value inequalities describe distances. Less-than inequalities produce inside intervals; greater-than inequalities produce outside intervals.

Equivalent exponential expressions use exponent rules, such as adding exponents when multiplying powers with the same base and multiplying exponents when raising a power to a power. Rewriting expressions with a common base is often the key step.

Equivalent expressions have the same value for every allowed value of the variable. Common strategies include expanding, factoring, combining like terms, applying exponent rules, and simplifying rational expressions while respecting restrictions on denominators.

Equivalent expressions have the same value for every allowed value of the variable. Common strategies include expanding, factoring, combining like terms, applying exponent rules, completing the square, and simplifying rational expressions while respecting denominator restrictions.

Equivalent expressions have the same value for every allowed value of the variable. Common strategies include expanding, factoring, combining like terms, applying exponent rules, completing the square, and simplifying rational expressions while respecting denominator restrictions.

Equivalent expressions have the same value for every allowed value of the variable. Common strategies include expanding, factoring, combining like terms, applying exponent rules, completing the square, and simplifying rational expressions while respecting denominator restrictions.

Equivalent expressions have the same value for every allowed value of the variable. Common strategies include expanding, factoring, combining like terms, applying exponent rules, completing the square, and simplifying rational expressions while respecting denominator restrictions.

Parameter questions for equivalent expressions often ask for missing coefficients or constants after expanding, factoring, completing the square, or simplifying rational expressions.

Equivalent-expression questions often require factoring before canceling, applying exponent rules, or simplifying radicals. Restrictions such as $x\ne4$ matter because canceling a factor does not make the original expression defined at the canceled value.

Polynomial expressions can often be simplified or rewritten by distributing, combining like terms, factoring out common factors, and recognizing special products such as differences of squares.

Quadratic expressions can be written in standard form, factored form, or vertex form. Moving between these forms reveals different information, such as zeros, coefficients, or the vertex.

Radical and rational expression questions require precise use of factoring, cancellation, exponent rules, and domain restrictions. Equivalent expressions must match the original expression on its allowed domain.

Function notation asks for outputs from specified inputs or for inputs that produce specified outputs. Transformations outside $f$ change outputs; transformations inside $f$ change inputs.

Exponential equations on the SAT often become linear equations after rewriting both sides with the same base. When bases match, the exponents can be set equal.

Nonlinear equations in one variable include quadratic, radical, rational, absolute value, exponential, and polynomial equations. Solving often requires factoring, taking square roots, isolating a radical, using exponent rules, or checking for restrictions and extraneous solutions.

Nonlinear equations in one variable can include quadratics, equations with radicals, rational equations, powers, and equations written in factored or transformed forms. Useful strategies include factoring, applying the zero product property, using inverse operations, checking for extraneous solutions, and interpreting parameters from the structure of an equation.

Nonlinear equations in one variable can include quadratics, equations with radicals, rational equations, powers, and equations written in factored or transformed forms. Useful strategies include factoring, applying the zero product property, using inverse operations, checking for extraneous solutions, and interpreting parameters from the structure of an equation.

Nonlinear equations in one variable can include quadratics, equations with radicals, rational equations, powers, and equations written in factored or transformed forms. Useful strategies include factoring, applying the zero product property, using inverse operations, checking for extraneous solutions, and interpreting parameters from the structure of an equation.

Nonlinear equations in one variable can include quadratics, equations with radicals, rational equations, powers, and equations written in factored or transformed forms.

Parameter questions for nonlinear equations often use roots, discriminants, squared forms, radical equations, or factored forms.

Polynomial equations are often solved by moving all terms to one side, factoring, and applying the zero product property. Some SAT-style items ask for a particular solution, a nonzero solution, or the number of real solutions shown by factored form.

Quadratic equations may be solved by factoring, taking square roots, or using the discriminant to reason about the number of real solutions. SAT-style items often ask for one solution, a parameter, or the number of real solutions rather than both roots.

Radical and rational equations are common sources of errors because restrictions and extraneous solutions must be checked. Squaring and clearing denominators can introduce invalid candidates.

Radical and rational equations require attention to restrictions. Rational equations can have excluded denominator values, and radical equations can produce extraneous solutions after squaring.

Exponential functions such as $f(x)=a(b)^x$ are interpreted using the initial value $a$ and the growth or decay factor $b$. Values greater than 1 indicate growth, while values between 0 and 1 indicate decay.

Nonlinear function questions may involve transformed functions, vertices, zeros, symmetry, evaluating composite-looking expressions, or comparing function features from different forms.

Nonlinear functions include quadratic, exponential, polynomial, radical, and rational functions. SAT-style questions often ask for function values, intercepts, maximum or minimum values, vertex information, transformations, or the meaning of parameters in context.

Nonlinear functions include quadratic, exponential, polynomial, radical, and rational functions. SAT-style questions often ask for function values, intercepts, maximum or minimum values, vertex information, transformations, domain restrictions, or the meaning of parameters in context.

Nonlinear functions include quadratic, exponential, polynomial, radical, and rational functions. SAT-style questions often ask for function values, intercepts, maximum or minimum values, vertex information, transformations, domain restrictions, or the meaning of parameters in context.

Nonlinear functions include quadratic, exponential, polynomial, radical, and rational functions. SAT-style questions often ask for function values, intercepts, maximum or minimum values, vertex information, transformations, domain restrictions, or the meaning of parameters in context.

Nonlinear functions include quadratic, exponential, polynomial, radical, and rational functions. SAT-style questions often ask for function values, intercepts, maximum or minimum values, vertex information, transformations, domain restrictions, or the meaning of parameters in context.

Parameter questions for nonlinear functions may use vertex form, domain restrictions, exponential factors, or input-output conditions.

Function questions involving radicals and rational expressions often ask for a value, an excluded input, or an endpoint of the domain. The key restrictions are that square-root radicands must be nonnegative and denominators cannot equal zero.

Polynomial function questions often use factored form to identify zeros and standard form to identify intercepts or evaluate function values. Parameter questions may ask for a coefficient or constant from a known zero or value.

Quadratic functions can be interpreted through vertex, standard, and factored forms. Vertex form gives the vertex and maximum or minimum value, while factored form gives the zeros and the axis of symmetry.

Radical and rational functions often ask for domain restrictions, excluded inputs, and endpoint values. Square-root radicands must be nonnegative, and denominators cannot equal zero.

Radical and rational equations often produce candidates that must be checked. Squaring can introduce extraneous solutions, and denominators cannot be zero.

Systems involving a line and a parabola can have zero, one, or two real solutions. Setting the equations equal usually creates a quadratic equation, and the discriminant determines the number of intersections.

Systems of equations in two variables can include linear and nonlinear equations. Common SAT-style systems pair a line with a parabola, a line with a circle-like equation, or a simple nonlinear relation with a linear relation.

Systems of equations in two variables can include linear and nonlinear equations. Common SAT-style systems pair a line with a parabola, a line with a circle-like equation, or a simple nonlinear relation with a linear relation.

Systems of equations in two variables can include linear and nonlinear equations. Common SAT-style systems pair a line with a parabola, a line with a circle-like equation, or a simple nonlinear relation with a linear relation.

Systems of equations in two variables can include linear and nonlinear equations. Common SAT-style systems pair a line with a parabola, a line with a circle-like equation, or a simple nonlinear relation with a linear relation.

Systems of equations in two variables can include linear and nonlinear equations. Common SAT-style systems pair a line with a parabola, a line with a circle-like equation, or a simple nonlinear relation with a linear relation.

Parameter questions for nonlinear systems often use a line-parabola system, root relationships, tangency, or identities involving sums and products.

A line-parabola system can be solved by setting the two expressions for $y$ equal. The resulting quadratic tells how many intersection points exist.