College Algebra 1314 HCC
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To find x -intercepts
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set y equal to zero and solve equation for x y=0
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To find y-intercepts
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set x equal to zero and solve the equation for y x=0
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A linear equation in one variable x is an equation ax+b=0
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where a and b are real numbers, with variable (a) does not equal zero
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Quadratic equation also called a second-degree polynomial equation in x.
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ax^2 + bx + c = 0 where a, b and c are real numbers with variable (a) does not equal zero.
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Zero-Factor Property
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only works for equations written in general form (in which the right side of the equation is zero). So all terms must be collected to one side before factoring.
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If ab=0, then a=0 or b=0
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Zero-Factor Property Write the left side of the general form of quadratic equation as the product of two linear factors. Then find the solution of the quadratic equation by setting each linear factor equal to zero.
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Extracting Square Roots
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The equation u^2=d, where d > 0 has exactly two solutions: u = √d and u= - √d. Because the two solutions differ in sign they can also be written as ± √d.
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\"u is equal to a plus or minus the square root of d\".
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Solving an equation of the form u^2= d without going through the steps of factoring is called extracting the square roots.
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Completing the Square if the left side is not factorable, you can not find its solutions unless you rewrite the equation by completing the square.
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You complete the square to solve a quadratic equation, you are just rewriting the equation so it can be solved by extracting the roots.
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x^2 + bx + (b/2)^2 = (x + b/2)^2
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to complete the square for the expression x^2 + bx, add (b/2)^2, which is the square of half the coefficient of x.
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x^2 + 2x-6 = 0
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x^2+2x=6 x^2 + 2x +1^2= 6+1^2 2x/2 = 1^2 (x+1)^2 = 7 x+1 = ± √7 x= -1 ± √7
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Quadratic Formula
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- b ± √ b² - 4ac 2a Negative b, plus or minus the square root of b squared minus 4ac, all divided by 2a.
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discriminant of quadratic formula is under the radical sign
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√ b² - 4ac
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Solutions - if the discriminant √ b² - 4ac is POSITIVE
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the quadratic equation has TWO distinct Real Solutions and its Graph has TWO x-intercepts
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Solutions - if the discriminant √ b² - 4ac is ZERO
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the quadratic equation has ONE Repeated Real Solution and its graph has ONE x-intercept
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Solutions - if the discriminant √ b² - 4ac is NEGATIVE
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the quadratic equation has NO Real Solutions and its graph has NO x-intercepts
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If discriminant of a quadratic equation is negative
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then its square root is imaginary (not a real number) and the Quadratic Formula has two complex solutions. Remember No Real Solution
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Real Numbers
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are represented graphically on the real number line. Numbers to the right of 0 are positive, and numbers to the left of 0 are negative. Are rational can also be fractions.
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symmetry
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graphs of equations can have symmetry with respect to one of the coordinate axes or with respect to the orgin.
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A graph is symmetric with respect to the x-axis if
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if whenever (x,y) is on the graph, (x, -y) is also on the graph
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A graph is symmetric with respect to the y-axis
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if, whenever (x,y) is on the graph (-x,y) is also on the graph
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A graph is symmetric with respect to the orgin
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if, whenever (x,y) is on the gaph (-x,-y) is also on the graph
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Algebraic Test for Symmetry ONE
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The graph of an equation is symmetric with respect to the x-axis if replacing y with -y yields an equivalent equation.
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Algebraic Test for Symmetry TWO
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The graph of an equation is symmetric with respect to the y-axis if replacing x with -x yields an equivalent equation.
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Algebraic Test for Symmetry THREE
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The graph of an equation is symmetric with respect to the orgin if replacing x with -x and y with -y yields an equivalent equation.
