Pts For Each Graph Is The Graph Symmetric With Respect To Th

1 10 Pts For Each Graph Is The Graph Symmetric With Respect To The

1. (10 pts) For each graph, is the graph symmetric with respect to the x-axis? y-axis? origin? (No explanation required. Just answer Yes or No to each question.)

• (a) Symmetric with respect to the x-axis? ____ y-axis? ____ origin? ____
• (b) Symmetric with respect to the x-axis? ____ y-axis? ____ origin? ____

2. (10 pts) Let f(x) = x³ + 2x (no explanation required)

• (a) State the zero(s) of the function. ________
• (b) Which of the following is true?
• A. f is an even function.
• B. f is an odd function.
• C. f is both even and odd.
• D. f is neither even nor odd.

3. (10 pts) Which of the following equations does the graph represent? Show work or explanation.

• 3. ______ A. B. C. D.

4. (10 pts) Look at the graph of the quadratic function and complete the table. [No explanations required.]

• Graph Fill in the blanks
• Equation: ________
• (a) State the vertex: ____________
• (b) State the range: _____________
• (c) State the interval on which the function is increasing: _____________
• (d) The graph represents which of the following equations? Choice:____ A. y = – x²+4x–3 B. y= –2 x² – x + 3 C. y= x² – 2 x – 3 D. y= x²+ 2 x – 3

5. (10 pts) Consider the points (–3, 7) and (3, –1).

• (a) Find the slope-intercept equation of the line passing through the two given points. Show work.
• (b) Graph the line you found in (a), either drawing it on the grid in the previous problem or generating the graph electronically and attaching it.
• (c) Compare your line for this problem, #4, with the line in the previous problem #3. Are the two lines parallel, perpendicular, or neither? (No explanation required — just state the answer.)

6. (10 pts) The number of guests g in a water park t hours after 9 am is given by g(t) = –32t² + 224t + 266 for 0 ≤ t ≤ 8 (t=0 corresponds to 9 am). Find and interpret the average rate of change of g over the interval [4,7]. Show work.

7. (10 pts) Luke wants to purchase custom-made mugs to advertise his business. Two companies offer different deals:

• Company A: Pay a design fee of $50.00, plus $4.50 per mug.
• Company B: Pay a design fee of $92.00, plus $3.90 per mug.
• (a) State a linear function f(x) that represents Company A's total charge for x mugs.
• (b) State a linear function g(x) that represents Company B's total charge for x mugs.
• (c) Luke wants 120 mugs, as cheaply as possible. Which company should he choose? Show work/explanation.
• (d) For what number of mugs is the total charge exactly the same for both companies? Show algebraic work/explanation.
• (e) Fill in the blanks: Company A is cheaper if ________ than ________ mugs are ordered.

8. (10 pts) A graph of y=f(x) follows. No formula for f is given.

• 8. _______ Which graph (A, B, C, or D) represents y=f(x₁)? Explain your choice.

9. (10 pts) Consider the graph of the piecewise function y=f(x) pictured below. (a) State the value of f(2). (b) State the x-intercept(s). (c) State the y-intercept(s). (d) State the domain of the piecewise function. (e) State the range of the piecewise function. (f) State the interval(s) where the function is increasing. (g) State the interval(s) where the function is decreasing.

10. (10 pts) Life expectancy at birth is the estimated lifespan of a baby born in a particular year. The regression line is y=0.2052x – 336.5, where x=birth year and y=life expectancy in years. The value of r² is 0.9809.

• (a) Use the regression line to estimate the life expectancy of a male born in 1970. Show work.
• (b) Predict the life expectancy for a baby born in 2020. Show work.
• (c) What is the slope of the regression line and what are its units? Interpret what the slope means in this context.
• (d) What is the value of the correlation coefficient r? Interpret its strength based on r.

Paper For Above instruction

The list of questions provided covers a broad spectrum of fundamental topics in algebra and precalculus, including symmetry of graphs, properties of functions, equations of lines, quadratic functions, analysis of graphs, and interpretation of regression data. Each problem assesses specific skills such as identifying symmetry, determining function characteristics, deriving equations from graphs, analyzing quadratic features, calculating slopes and graphing linear equations, interpreting average rates of change, modeling costs with linear functions, recognizing transformations of functions, describing properties of piecewise functions, and understanding regression analysis for real-world data. The aim is to demonstrate proficiency in these core mathematical concepts through concise answers and explanations, illustrating understanding of the underlying principles and applications.

1. Symmetry of graphs with respect to axes and origins is a key concept in understanding the behavior of functions and their visual representations. For example, graphs symmetric with respect to the x-axis are reflected over the x-axis, indicating that f(x) and –f(x) produce the same graph. Symmetry about the y-axis indicates the function is even, satisfying f(–x) = f(x). Symmetry with respect to the origin indicates the function is odd, satisfying f(–x) = –f(x). Determining these symmetries involves analyzing the algebraic form of functions or visually inspecting the graph. Such properties aid in sketching graphs and understanding the nature of the functions.

2. Determining the zeros of a function like f(x) = x³ + 2x involves solving algebraic equations, here, factoring or using roots. Recognizing whether the function is even, odd, or neither involves testing the symmetry properties: an even function satisfies f(–x) = f(x), an odd function satisfies f(–x) = –f(x). For the given cubic, testing these properties helps classify the function, which is useful in understanding its symmetry and graph behavior.

3. Identifying which equation a graph represents involves comparing the visual features of the graph with the expected shape of functions, such as linear, quadratic, cubic, etc., and matching key points, intercepts, vertex, and overall shape. Showing work involves analyzing the slope, intercepts, and features like concavity or symmetry to determine the best match among given options.

4. Analyzing quadratic functions involves identifying the vertex form or standard form to determine the vertex, range, and increasing/decreasing intervals. The vertex indicates the maximum or minimum point, and the parabola opens upward or downward depending on the leading coefficient. Recognizing which quadratic equation corresponds to a given graph requires understanding these features and how algebraic forms reflect the graph's geometry.

5. Calculating the equation of a line given two points involves using the slope formula: m = (y₂ – y₁) / (x₂ – x₁), then using the point-slope form or slope-intercept form to derive the equation. Graphical comparison of two lines allows for classification of their relationship: parallel, perpendicular, or neither, based on their slopes.

6. Calculating the average rate of change of a function over an interval involves the difference quotient: [g(b) – g(a)] / (b – a), providing insight into the overall increase or decrease of the function over that interval. Interpreting this in context clarifies the average increase in guests in the water park per hour over specific times.

7. Modeling costs with linear functions involves understanding fixed costs (intercepts) and variable costs per unit. Comparing two linear models for different deals allows decision-making regarding the most economical choice over a certain number of mugs. Setting the functions equal solves for the point where costs are the same, guiding the choice for different quantities.

8. Recognizing transformed functions, such as y=f(x₁), from graphs involves understanding how shifts, stretches, and compressions affect the graph. Identifying which graph corresponds to a particular transformation necessitates analyzing features like domain, range, and intercepts, relative to the original graph.

9. For piecewise functions, key aspects include evaluating the function at specific points, identifying intercepts, determining the domain (set of x-values), range (set of y-values), and analyzing the function's increasing or decreasing behavior over various intervals. These are essential in understanding the overall behavior of the function.

10. In regression analysis, estimating the value of a dependent variable based on the regression line involves plugging the independent variable's value into the equation. The slope indicates the rate of change per unit increase in x, and the correlation coefficient measures the strength of the linear relationship, with values close to 1 indicating a very strong correlation. Interpreting these helps understand the data's real-world implications, such as changes in life expectancy over years.

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