Course Review: Mixed Practice on Notation, Evaluation, Representations, and Transformations

How to Use This Review Chapter

This chapter is a consolidation workspace. You will practice recognizing function notation, evaluating expressions, reasoning about domain/range, connecting tables/graphs/formulas, and describing transformations by both sketching and writing equations. For each section: (1) write your work clearly, (2) label key points (intercepts, vertices, asymptotes, endpoints), and (3) state domain/range using interval notation when possible.

1) Diagnostic Warm-Up (Notation, Evaluation, Domain/Range)

Do these quickly to identify what needs review. Show enough work to verify each answer.

A. Notation and meaning

• 1. If f(2)=9, what does this statement mean in words?
• 2. Explain the difference between f(3) and f(x).
• 3. If g(x)=f(x)+2, describe how the outputs of g compare to the outputs of f for the same input.

B. Evaluate (substitute carefully)

• 4. Let f(x)=2x^2-3x+1. Find f(-2).
• 5. Let h(x)=|x-5|. Find h(1) and h(9).
• 6. Let p(x)=\frac{1}{x-4}. Find p(6). State any input that is not allowed.
• 7. Let r(x)=\sqrt{x+3}. Find r(13). State the domain of r.

C. Domain and range reasoning (no graph needed)

• 8. For m(x)=\sqrt{7-x}, state the domain and range.
• 9. For q(x)=\frac{2}{x+1}-5, state the domain and range.
• 10. For s(x)=-(x-2)^2+4, state the domain and range.

2) Multi-Representation Tasks (Match Table, Graph Features, and Formula)

In each task, you will match items that represent the same function. Use specific evidence: constant rate of change, symmetry, vertex location, intercepts, asymptotes, or endpoint behavior.

Task 2.1: Match formulas to tables

Match each formula (A–D) to one table (1–4). Justify with one sentence of evidence (pattern, symmetry, or key values).

Formulas

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• A. f(x)=2x+1
• B. g(x)=(x-1)^2
• C. h(x)=|x+2|
• D. k(x)=\frac{1}{x}

Tables

x: -2  -1   0   1   2
y: -3  -1   1   3   5

x: -1   0   1   2   3
y:  4   1   0   1   4

x: -4  -3  -2  -1   0
y:  2   1   0   1   2

x: -2  -1   1   2
y:-1/2 -1   1  1/2

Task 2.2: Match formulas to graph descriptions

Match each formula (E–H) to one graph description (i–iv). Use features (vertex, asymptotes, intercepts, end behavior).

Formulas

• E. y=(x+3)^2-2
• F. y=-|x-1|+4
• G. y=\sqrt{x-5}+1
• H. y=\frac{2}{x+1}

Graph descriptions

• i. A parabola opening up with vertex at (-3,-2).
• ii. A V-shape opening down with peak at (1,4).
• iii. A square-root curve starting at (5,1) and increasing to the right.
• iv. A reciprocal-type curve with vertical asymptote x=-1 and horizontal asymptote y=0.

Task 2.3: Build a table from a graph feature list

A function has these features: it is linear, crosses the y-axis at (0,-2), and rises 3 units for every 1 unit to the right. Create (a) an equation, and (b) a 5-row table of values including x=-1,0,1,2,3.

3) Transformation Sequences (Sketch from Parent + Write the Equation)

For each problem: (1) name the parent function family, (2) list transformations in order, (3) sketch by plotting key points/asymptotes/endpoints, and (4) write the final equation. Use function notation when helpful, e.g., y=a f(b(x-h))+k.

Transformation Toolkit (quick checklist)

• Horizontal shift: f(x-h) shifts right h; f(x+h) shifts left h.
• Vertical shift: f(x)+k shifts up k; f(x)-k shifts down k.
• Vertical scale/reflection: a f(x) stretches by |a|; if a<0 reflect across x-axis.
• Horizontal scale/reflection: f(bx) compresses by |b| if |b|>1; stretches if 0<|b|<1; if b<0 reflect across y-axis.

Task 3.1: Quadratic sequence (from y=x^2)

Start with y=x^2. Apply transformations in order: (1) shift right 2, (2) reflect across the x-axis, (3) vertical stretch by factor 3, (4) shift up 5.

• a) Write the final equation.
• b) List the vertex and whether the parabola opens up or down.
• c) State the range.

Task 3.2: Absolute value sequence (from y=|x|)

Start with y=|x|. Apply: (1) shift left 4, (2) vertical compression by factor 1/2, (3) shift down 3.

• a) Write the final equation.
• b) Give the vertex and two additional points you would plot.
• c) State domain and range.

Task 3.3: Radical sequence (from y=\sqrt{x})

Start with y=\sqrt{x}. Apply: (1) shift right 1, (2) reflect across the x-axis, (3) shift up 2.

• a) Write the final equation.
• b) Identify the new starting point (endpoint) and the direction the curve moves.
• c) State domain and range.

Task 3.4: Reciprocal sequence (from y=\frac{1}{x})

Start with y=\frac{1}{x}. Apply: (1) shift left 2, (2) vertical stretch by factor 4, (3) shift down 1.

• a) Write the final equation.
• b) State the vertical and horizontal asymptotes.
• c) State domain and range.

Task 3.5: From an equation to transformations (reverse engineering)

For each function, identify the parent function and describe a clear transformation sequence.

• a) y=(x+1)^2-9
• b) y=2|x-3|+1
• c) y=-\sqrt{x+4}
• d) y=\frac{-3}{x-2}+5

4) Reflection Prompts (Explain Recognition Using Features)

Answer in complete sentences. Use specific features (rate of change, symmetry, vertex, asymptotes, endpoint, intercept behavior).

• 1. When you see a graph with a single endpoint and then a curve that increases slowly, what family do you suspect first, and what feature supports that?
• 2. What two features most quickly distinguish an absolute value graph from a quadratic graph?
• 3. If a graph has a vertical asymptote at x=4 and a horizontal asymptote at y=-2, what general form of equation do you look for?
• 4. Describe how you can tell from an equation whether a horizontal shift is left or right without graphing.
• 5. In your own words, explain how the sign and size of a coefficient outside a function (like a f(x)) affects the graph.

Final Mixed Problem Set (All Skills Integrated)

Section A: Notation, evaluation, and interpretation

• A1. If f(x)=x^2-4x, compute f(0), f(4), and f(-1).
• A2. If g(x)=\frac{x+2}{x-3}, evaluate g(5) and state the excluded input(s).
• A3. A function p satisfies p(2)=-1 and p(6)=7. Interpret these as points on a graph.

Section B: Domain and range

• B1. Find the domain of y=\sqrt{2x-8}.
• B2. Find the domain and range of y=\frac{1}{x+5}.
• B3. Find the range of y=-(x-3)^2+10.

Section C: Match representations

Match each equation to the correct key feature list.

Equations

• C1. y=|x+1|-2
• C2. y=(x-2)^2+1
• C3. y=\frac{2}{x-4}
• C4. y=\sqrt{x+3}

Feature lists

• F1. Vertex at (-1,-2), V-shape opening up.
• F2. Vertex at (2,1), parabola opening up.
• F3. Vertical asymptote x=4, horizontal asymptote y=0.
• F4. Endpoint at (-3,0), defined only for x\ge -3.

Section D: Transformations (write equation + key features)

• D1. Start with y=|x|. Reflect across the x-axis, shift right 5, then shift up 2. Write the equation and state the vertex.
• D2. Start with y=x^2. Shift left 3 and down 4, then apply a vertical stretch by 2. Write the equation and state the vertex.
• D3. Start with y=\frac{1}{x}. Reflect across the y-axis, shift up 3, then shift left 1. Write the equation and state both asymptotes.
• D4. Start with y=\sqrt{x}. Compress horizontally by factor 2 (i.e., make it steeper), then shift right 4. Write the equation and state the endpoint.

Section E: Short explanation items

• E1. A function has a vertex at (-2,6) and opens downward. Write one possible equation and explain how you know it opens downward.
• E2. Explain how you can confirm from a table whether a function is likely linear or not (without graphing).

Answer Key Plan (Final Answers + Brief Transformation Reasoning)

Use this plan to check your work: record (1) final numeric answers, (2) domain/range in interval notation, and (3) for transformations, a one-line identification of each change (shift/reflection/scale) tied to the equation structure.

Warm-Up Answers

• 1. Input 2 produces output 9 (point (2,9) is on the graph).
• 2. f(3) is a number (the output at input 3); f(x) is the output expression for a general input x.
• 3. g outputs are always 2 more than f at the same input (vertical shift up 2).
• 4. f(-2)=2(4)-3(-2)+1=8+6+1=15.
• 5. h(1)=|1-5|=4, h(9)=|9-5|=4.
• 6. p(6)=1/(6-4)=1/2; excluded input: x\ne 4.
• 7. r(13)=\sqrt{16}=4; domain: x\ge -3 i.e., [-3,\infty).
• 8. Domain: 7-x\ge 0 \Rightarrow x\le 7 so (-\infty,7]; range: [0,\infty).
• 9. Domain: x\ne -1; range: y\ne -5 (reciprocal shift down 5).
• 10. Domain: (-\infty,\infty); range: maximum at 4 so (-\infty,4].

Multi-Representation Answers

• 2.1 Matches: A→Table 1 (constant +2 change), B→Table 2 (symmetric squares around x=1), C→Table 3 (V with vertex at x=-2), D→Table 4 (reciprocal pairs).
• 2.2 Matches: E→i (vertex (-3,-2)), F→ii (downward V peak (1,4)), G→iii (endpoint (5,1)), H→iv (asymptotes x=-1, y=0).
• 2.3 Equation: y=3x-2; table: x=-1,0,1,2,3 gives y=-5,-2,1,4,7.

Transformation Task Answers

• 3.1 y=-3(x-2)^2+5; vertex (2,5); opens down; range (-\infty,5]. Reasoning: (x-2) right 2, negative reflects, 3 stretches, +5 up.
• 3.2 y=\tfrac12|x+4|-3; vertex (-4,-3); example points: at x=-3, y=-2.5; at x=-5, y=-2.5; domain (-\infty,\infty), range [-3,\infty).
• 3.3 y=-\sqrt{x-1}+2; endpoint (1,2); moves rightward while decreasing; domain [1,\infty), range (-\infty,2].
• 3.4 y=\frac{4}{x+2}-1; asymptotes x=-2, y=-1; domain x\ne -2, range y\ne -1.
• 3.5 a) Parent x^2; left 1, down 9. b) Parent |x|; right 3, vertical stretch 2, up 1. c) Parent \sqrt{x}; left 4, reflect across x-axis. d) Parent 1/x; right 2, reflect+stretch by 3 (negative outside), up 5.

Final Mixed Problem Set Answers

Section A

• A1. f(0)=0, f(4)=16-16=0, f(-1)=1+4=5.
• A2. g(5)=(7)/(2)=7/2; excluded input: x\ne 3.
• A3. Points (2,-1) and (6,7) lie on the graph of p.

Section B

• B1. 2x-8\ge 0 \Rightarrow x\ge 4; domain [4,\infty).
• B2. Domain x\ne -5; range y\ne 0.
• B3. Maximum 10 at x=3; range (-\infty,10].

Section C

• C1→F1, C2→F2, C3→F3, C4→F4.

Section D

• D1. y=-|x-5|+2; vertex (5,2). Reasoning: negative reflects, (x-5) right 5, +2 up.
• D2. y=2(x+3)^2-4; vertex (-3,-4). Reasoning: (x+3) left 3, -4 down, factor 2 stretch.
• D3. y=\frac{1}{-(x+1)}+3=-\frac{1}{x+1}+3; asymptotes x=-1, y=3. Reasoning: reflect in y-axis via x\to -x, then shifts.
• D4. y=\sqrt{2(x-4)}; endpoint (4,0). Reasoning: f(2x) is horizontal compression by 2; then right 4 gives 2(x-4) inside.

Section E

• E1. Example: y=-(x+2)^2+6; opens downward because the coefficient on the squared term is negative (reflection across x-axis).
• E2. Linear tables show constant first differences in y when x increases by equal steps; non-constant differences suggest non-linear behavior (e.g., quadratic has constant second differences).