AoPS Intermediate Algebra Chapter 1-5

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Questions and Answers

What is a consistent system?

A system with at least one solution (correct)

A system with no solution

A system where all equations are independent

A system with multiple solutions

What characterizes an inconsistent system?

No solution exists (correct)

All equations are linear combinations of each other

Exactly one solution exists

At least one solution exists

What is a dependent system?

All equations are independent

At least one equation is a linear combination of the others (correct)

There are multiple solutions

No equation is a linear combination of the others

What defines an independent system?

No equation is a linear combination of the others Signup and view all the answers

What is an extraneous solution?

A solution to the squared equation that does not satisfy the original equation. Signup and view all the answers

What is the domain of a function?

All the values we can input to the function to get an output. Signup and view all the answers

What is the range of a function?

All the values that can possibly come out of the function. Signup and view all the answers

What does the vertical line test determine?

A curve is the graph of a function if every vertical line intersects it at no more than one point. Signup and view all the answers

What are inverse functions?

Functions f and g are inverse functions if g(f(x))=x and f(g(x))=x for all values of x in their domains. Signup and view all the answers

Describe the graph of y=f(x)+k.

The graph of y=f(x) vertically shifts upward by k units. Signup and view all the answers

Describe the graph of y=f(x+k).

The graph of y=f(x) is horizontally shifted to the left by k units. Signup and view all the answers

What is the reflection of y=f(x) over the x-axis?

The graph of y=-f(x). Signup and view all the answers

What is the reflection of y=f(x) over the y-axis?

The graph of y=f(-x). Signup and view all the answers

What does y=kf(x) represent?

It results from scaling the graph of y=f(x) vertically by a factor of k. Signup and view all the answers

What does y=f(kx) represent?

It results from scaling the graph of y=f(x) horizontally by a factor of 1/k. Signup and view all the answers

What is the composition of two functions?

Putting the output from the first function into the second (e.g., f(g(x))). Signup and view all the answers

What is the conjugate of a+bi?

a-bi. Signup and view all the answers

What is the magnitude of a + bi?

√(a²+b²). Signup and view all the answers

What is the conjugate of (z+w)?

Conjugate of z + conjugate of w. Signup and view all the answers

What is the discriminant of a quadratic?

b²-4ac. Signup and view all the answers

If the discriminant is 0, there is one root.

True Signup and view all the answers

If the discriminant is positive, the roots are non-real.

False Signup and view all the answers

If the discriminant is negative, the roots are real.

False Signup and view all the answers

What are Vieta's formulas?

r+s=-b/a and rs=c/a. Signup and view all the answers

What is a locus?

A set of points that satisfy certain given conditions. Signup and view all the answers

What is a parabola?

The locus of points equidistant from a given line (the directrix) and a point (the focus). Signup and view all the answers

What is the axis of symmetry?

The line through the focus that is perpendicular to the directrix. Signup and view all the answers

What is a circle?

The locus of points in a plane a specified distance (the radius) from a given point (the center). Signup and view all the answers

What is the standard form of a circle?

(x-h)² + (y-k)² = r². Signup and view all the answers

What is an ellipse?

The locus of points P in a plane such that PF1 + PF2 is a given constant. Signup and view all the answers

What is the standard form of an ellipse?

(x-h)²/a² + (y-k)²/b² = 1. Signup and view all the answers

What is the major axis of an ellipse?

The line segment joining the vertices of an ellipse. Signup and view all the answers

What is the minor axis of an ellipse?

The short axis of an ellipse, perpendicular to the major axis. Signup and view all the answers

What is the distance between the foci of an ellipse?

(major axis length)² = (minor axis length)² + (distance between the foci)². Signup and view all the answers

What are the asymptotes of a hyperbola?

The lines that the hyperbola approaches but doesn't intersect (y-k)/a = ±(x-h)/b. Signup and view all the answers

Study Notes

Systems of Equations

A consistent system has at least one solution.
An inconsistent system has no solutions.
A dependent system has equations that are linear combinations of each other.
An independent system has equations that are not linear combinations of one another.

Solutions and Functions

An extraneous solution is a valid solution for a squared equation but doesn't satisfy the original equation.
The domain of a function includes all possible input values that produce an output.
The range of a function encompasses all possible output values.

Graphing Functions

The vertical line test determines if a curve represents a function by checking if vertical lines intersect it at most once.
Functions f and g are mutual inverses if ( g(f(x)) = x ) and ( f(g(x)) = x ) for all ( x ) in their domain.

Graph Transformations

The graph of ( y = f(x) + k ) shifts vertically by ( k ) units.
The graph of ( y = f(x + k) ) shifts horizontally to the left by ( k ) units.
The graph of ( y = -f(x) ) reflects over the x-axis.
The graph of ( y = f(-x) ) reflects over the y-axis.
The graph of ( y = kf(x) ) scales vertically by a factor of ( k ).
The graph of ( y = f(kx) ) scales horizontally by a factor of ( 1/k ).

Functions and Magnitudes

The composition of two functions involves substituting the output of the first function into the second (e.g., ( f(g(x)) )).
The conjugate of ( a + bi ) is ( a - bi ).
The magnitude of ( a + bi ) is ( \sqrt{a^2 + b^2} ).

Quadratic Equations

The discriminant of a quadratic ( ax^2 + bx + c ) is given by ( b^2 - 4ac ).
If the discriminant equals 0, there is one root.
If the discriminant is positive, the roots are real.
If the discriminant is negative, the roots form a non-real conjugate pair.
Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots: ( r+s = -b/a ) and ( rs = c/a ).

Geometric Structures

A locus is a set of points satisfying specific conditions.
A parabola is defined as points equidistant from a directrix and a focus.
The axis of symmetry in a parabola is the line through the focus that is perpendicular to the directrix.
A circle is the set of points at a fixed distance (radius) from a center.
The standard form of a circle is expressed as ( (x-h)^2 + (y-k)^2 = r^2 ).
An ellipse is defined where the sum of distances from two foci remains constant.
The standard form of an ellipse is ( (x-h)^2/a^2 + (y-k)^2/b^2 = 1 ).
The major axis connects the vertices of an ellipse.
The minor axis is shorter and perpendicular to the major axis.
The distance between foci can be derived from the major and minor axis lengths using the equation: ( (major\ axis\ length)^2 = (minor\ axis\ length)^2 + (distance\ between\ the\ foci)^2 ).

Hyperbola Properties

The asymptotes of a hyperbola are lines approached by the hyperbola but never intersected, described by the equation ( (y-k)/a = \pm (x-h)/b ).

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Description

Test your knowledge of key terms in Intermediate Algebra from Chapters 1 to 5. This quiz covers essential definitions like consistent and inconsistent systems, as well as dependent and independent systems. Ideal for students looking to reinforce their understanding of algebraic concepts.