H-Pre-Calculus Targets Chapter Section. Sketch and analyze graphs of quadratic functions.. I can write quadratic functions in standard form and use the results to sketch graphs of the function. Identify the vertex and x-intercepts and sketch a graph of each of the following. f ( x ) = x + x + f ( x ) = x - x + Convert each of the following to standard form and identify the vertex. g( x ) = x - 6x - h( x ) = x - 0x - Write the standard for of the quadratic function that has the indicated vertex and whose graph passes through the given point. e. vertex: ( ) through ( ) f. vertex: (- -6) through ( -) Find two quadratic functions one that opens upward and one that opens downward whose graphs have the given x- intercepts. (There are many correct answers.) g. (- 0) & ( 0) h. ( 0) and ( 0). I can find minimum and maximum values of functions in real-life applications. Find two positive real numbers whose product is a maximum and whose sum is 90. Find two positive real numbers whose sum of the first number and twice the second number is and whose product is a maximum. Find the point on the curve of y = x that is closest to the point ( 0). Suppose that the revenue generated by selling x units of a certain commodity is given by the equation R = -0.x + 00x. Assume that R is in dollars. What is the maximum revenue possible in this situation? How many units must be sold to maximize the revenue? See worksheets & for additional examples of this target. Section. Sketch and analyze graphs of polynomial functions.. I can use transformations to sketch graphs of polynomial functions. Compare the graphs of Compare the graphs of f ( x) f ( x) = x f x ( x ) ( ) = + and f x ( x ) = x f ( x) ( ) x ( ) = + = + and f x ( x ) ( ) = +. I can use the Leading Coefficient Test to determine the end behavior of graphs of polynomial functions. ( ) = + 7 + ( ) = 7 6 ( ) = + 7 + ( ) = 7 Left-side behavior Right-side behavior.
. I can find and use zeros of polynomial functions as sketching aids. e. f ( x) x x 0x = f ( x) x x x = + f ( x) = ( x ) ( x + ) f. f ( x) = x + x + 0x f ( x) = x + x + 0x ( ) = ( + ) ( ) ( ) 6. I can use the Intermediate Value Theorem to help locate intervals of length that contain zeros of polynomial functions. f ( x) = x + x + x ( ) = 07 + 8 + ( ) = 8 + + Section. Use long division and synthetic division to divide polynomials by other polynomials and determine the numbers of rational and real zeros of polynomial functions and find the zeros. 7. I can use long division to divide polynomials by other polynomials. (x + x + 7x + ) (x + ) (x + x + 7x + ) (x - ) (x + x - x + x - ) (x + x - ) (x + x + ) (x x + ) 8. I can use synthetic division to divide polynomials by binomials of the form (x k) (x - x - x + 7x + ) (x - ) (x - x + 7x + ) (x + ) (x - x - 8x + x - ) (x - 6) 9. I can use the Remainder and Factor Theorems. Use the Remainder Theorem to evaluate each polynomial function at the given x value. f(x) = x - 7x + 8x + ) at x = f(x) = x - x + x - x + at x = - Find all real zeros of the polynomial function given some factor(s). f(x) = x + x - 9x + 6 factors (x + ) and (x ) f(x) = 8x - x - 7x 0x + factors (x + ) and (x ) 0. I can use the Rational Zero Test to determine possible rational zeros of polynomial functions. List all possible rational zeros f(x) = x + 7x - 9x 6 f(x) = x 7x + 8x + 9x - x + f(x) = 6x - 8x - 9x +. I can use Descartes Rule of Signs and the Upper and Lower Bound Rules to find zeros of polynomials. Use Descartes Rule of Signs to determine the possible numbers of positive and negative real zeros. f(x) = x - 7x + 8x + x - f(x) = x + x x x - x + f(x) = x 7 x 6 + x 7x x + x + x Verify the Upper and Lower Bounds f(x) = x + x + x Upper Bound: Lower Bound: -6 e. f(x) = x - x - 8x + x + 6 Upper Bound: Lower Bound: -
Section. Perform operations with complex numbers and plot complex numbers in the complex plane.. I can use the imaginary unit i to write complex numbers in standard form. 8 + 9 i + i i + i 7. I can add subtract and multiply complex numbers. i( i) i( + i) i( i)( + i) ( ) ( ) i + i ( i) ( + i). I can use complex numbers to write the quotient of two complex numbers in standard form. i + i i 7 i. I can plot complex numbers in the complex plane. i i Section. Use applications of the Fundamental Theorem of Algebra 7. I can find all zeros of polynomial functions including complex zeros. f ( x) = x x + 0x h( x) = x x 9x g x x x x x ( ) = + 7 + 0 ( ) = 9 + 0 8. I can find conjugate pairs of complex zeros. Write a polynomial function the real coefficients that has the given zeros. i i + i + i 9. I can find zeros of polynomials by factoring. f x x x ( ) = + 8 f ( x) = x x x + x 0 if one factor is x f ( x) = x x + x + 8x 6 if one factor is x x Section.6 &.7 Graphs of rational functions. 0. I can determine the domains of rational functions.. I can find any holes in the graph of rational functions.. I can find the vertical horizontal and slant asymptotes of rational functions.. I can find the x- and y-intercepts of rational functions.. I can determine if the graph of a rational function crosses the horizontal or slant asymptote. y = (x+ )(x - ) (x - )(x+) y = x - x - 0 x + e. y = x + x - ( x - )( x + )( x + )( x ) f. y = x + ( x + )( x )( x )( x + ) ( x + )( x )( x + ) (x + 6)(x + ) g. y = h. y = ( x )( x + ) (x - ) y = x + 7x + x + x +
H-Pre-Calculus Targets Chapter Answers vertex (- 0) x-int. (-0) 7 x-int. none vertex ( ) 8 g( x) = x 6 vertex ( -6) ( ) h( x) = x 0 vertex ( -0) ( ) f ( x) = x + 7 e. ( ) 9 f. ( ) 9 f ( x) = x + 6 9 g. many answers are possible f ( x) = x + x 0 f ( x) = x x + 0 h. many answers are possible f ( x) = x x + f x x x & 6 & 8 0 ( ) ( ) = + $8. 66.66units 000 00 a & use your calculator left: right: left: right: + left: + right: + left: + right: 0 zeros { } zeros { 0 } zeros { 0 } zeros { 0 } e. zeros { } f. zeros { 0 } 6 ( 0 ) 6 at - ( 0 ) ( ) Note: other in (- 0) 6 (- -) (- 0) (0 ) ( ) 7 x x + x + 6 7 x + 7x + + x x x + + + x + x 7 7 x x + x + + x x+ 8 8 8 9 9 6 x x x x 9 x x + + 6 x + x x + x 6 7 x 9 { } 9 { } 0 ± ± ± ± 6 0 ± ± ± ± ± ± 0 ± ± ± ± ± 6 ± ± ± ± ± ± ± 6 positive: or negative: positive: or 0 negative: or positive: or or negative: or 0 d & e. all bounds are verifiable 8 + 7i + i + i 8 + 66i 6i i i 9 + i a & plot on real & complex axis 0.869.0 7. { } complex (unable to find) {.0 ±.0i} 7 { } ± i 7 { 6.8} complex (unable to find){ 0.96 ±.6i} ± i 7 7 { } f x = x + x + 9x + 00x + 00 8 ( ) f x = x + 8x + 0x x + 6 8 ( ) f x = x + 9x + 8x + 78x 6 8 ( ) f x = x + x 0x + 9 8 ( )
9 { ± i ± } ± 9 { ± } 9 { } ± 7 ± i 7 0 VA. x=- Holes: ( ) x-int. (- 0) y-int. (0 ) x : x R x Domain: { } 0 VA. none HA none Holes: (- -7) x-int. ( 0) y-int. (0 -) Domain: ( + ) Range: ( + ) 0 VA. x= Holes: ( ) x-int. (- 0) y-int. (0 ) x } 0e. VA. none HA y = Holes: none ± x-int. ( 0 ) y-int. (0 ) Domain: ( + ) y } 0f. VA. x = - x = Holes: ( 0) x-int. (- 0) y-int. (0 6) AX ( ) x } 0g. VA. x = - HA none SA y = x + Holes: ( ) x-int. (- 0) & ( 0) y-int. ( 0 ) AX none x } Range: ( + ) 0h. VA. x = - Holes: none x-int. (- 0) & (-6 0) y-int. (0 ) AX ( ) x }