Teaching Flashcards

01. Explain Slope

Teach: What does slope tell us?

1. Slope describes a line's steepness.
2. It is change in y over change in x.
3. Think: rise divided by run.
4. In y = 2x + 3, the slope is 2.
5. Move right 1, then move up 2.
6. A positive slope rises to the right.
7. A negative slope falls to the right.
8. A horizontal line has slope zero.
9. Try: Describe a slope of -3.

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02. Find the Intercepts

Teach: How do we find intercepts?

1. Intercepts are axis crossings.
2. At the x-axis, y equals zero.
3. At the y-axis, x equals zero.
4. Example: x + y = 6.
5. Set y = 0: x = 6.
6. The x-intercept is (6, 0).
7. Set x = 0: y = 6.
8. The y-intercept is (0, 6).
9. Try: Find both for x + y = 4.

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03. Choose the Right Rule

Teach: How does piecewise work?

1. Different inputs use different rules.
2. Read the input condition first.
3. Use f(x) = 2x when x < 0.
4. Use f(x) = 3 when x >= 0.
5. For x = -2, use the first rule.
6. So f(-2) = 2(-2) = -4.
7. For x = 1, use the second rule.
8. So f(1) = 3, not 2.
9. Try: Explain why f(0) = 3.

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04. Open and Closed Dots

Teach: What do endpoint dots mean?

1. An open dot excludes that point.
2. A closed dot includes that point.
3. Use an open dot for < or >.
4. Use a closed dot for <= or >=.
5. Graph each rule only where allowed.
6. For y = 2x when x < 0:
7. Draw only to the left of x = 0.
8. Leave an open dot at (0, 0).
9. Try: What changes for x <= 0?

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05. Explain a Step Function

Teach: Why does the graph have steps?

1. Each interval has one fixed output.
2. That makes each piece horizontal.
3. The output can jump at a boundary.
4. Think of fees by parking time.
5. One time bracket has one price.
6. The next bracket may cost more.
7. Check which bracket owns an endpoint.
8. Open dots exclude; closed include.
9. Try: Describe another step model.

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06. Round Down with Floor

Teach: What does floor(x) mean?

1. Choose the greatest integer <= x.
2. This means round down, not to zero.
3. floor(3.8) = 3.
4. floor(3) = 3.
5. floor(-0.2) = -1, not 0.
6. floor(-6.35) = -7, not -6.
7. On 2 <= x < 3, floor(x) = 2.
8. Include 2; exclude 3 on that step.
9. Try: Explain floor(-2.4).

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07. Distance Makes a V

Teach: What is absolute value?

1. It is distance from zero.
2. Distance cannot be negative.
3. So |-5| = 5 and |5| = 5.
4. The graph of y = |x| is a V.
5. Its vertex is (0, 0).
6. Opposite inputs give equal outputs.
7. The domain is all real numbers.
8. The range is y >= 0.
9. Try: Why is |-8| not -8?

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08. Solve Absolute Value

Teach: Why can there be two answers?

1. Two numbers can share a distance.
2. Example: |x - 1| = 4.
3. Write x - 1 = 4 or x - 1 = -4.
4. Solve to get x = 5 or x = -3.
5. Both are 4 units away from 1.
6. Check both in the original equation.
7. Isolate the absolute value first.
8. |x| = -4 has no real solution.
9. Try: Solve |x| = 3 and explain.

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09. Recognize Parent Shapes

Teach: What is a parent function?

1. It is a family's basic function.
2. Transformations build related graphs.
3. y = x is a straight diagonal line.
4. y = |x| makes a V shape.
5. y = x^2 makes a curved parabola.
6. y = 3 makes a horizontal line.
7. Shape helps you choose a model.
8. Do not confuse a V with a curve.
9. Try: Sketch y = |x| and y = x^2.

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10. Domain Versus Range

Teach: Compare domain and range.

1. Domain means allowed inputs.
2. Range means possible outputs.
3. For y = x^2, any real x works.
4. But squaring never gives a negative.
5. Domain: all real numbers.
6. Range: y >= 0.
7. For y = 3, any real x still works.
8. Its range contains only 3.
9. Try: Give the range of y = |x|.

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11. Read Graph Shifts

Teach: How do we read a translation?

1. A translation moves the whole graph.
2. It does not change the shape.
3. In g(x) = f(x - h) + k:
4. h moves right; k moves up.
5. Negative h moves left.
6. Negative k moves down.
7. Example: y = (x - 3)^2 + 2.
8. Shift y = x^2 right 3 and up 2.
9. Try: Describe y = (x + 4)^2 - 1.

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12. Move a Known Point

Teach: How do graph points shift?

1. Move every point the same distance.
2. Example: y = |x - 2| + 3.
3. Start with parent y = |x|.
4. Move right 2 and up 3.
5. The vertex (0, 0) becomes (2, 3).
6. The point (1, 1) becomes (3, 4).
7. Check: |3 - 2| + 3 = 4.
8. The V keeps the same shape.
9. Try: Where does (-1, 1) move?

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13. Stretch Vertically

Teach: What does a*f(x) change?

1. It multiplies each output by a.
2. The x-coordinate stays the same.
3. For y = 3x^2, multiply heights by 3.
4. (1, 1) becomes (1, 3).
5. This is a vertical stretch by 3.
6. For y = (1/2)x^2, halve heights.
7. That is a vertical compression.
8. A negative a also flips over x-axis.
9. Try: Describe y = -2x^2.

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14. Scale Horizontally

Teach: What does f(b*x) change?

1. It divides x-coordinates by b.
2. For b = 2, widths are halved.
3. This is a horizontal compression.
4. For b = 1/2, widths are doubled.
5. This is a horizontal stretch.
6. In y = |2x|, start from y = |x|.
7. (2, 2) becomes (1, 2).
8. A negative b also flips over y-axis.
9. Try: Describe y = |x/3|.

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15. Combine Flip and Shift

Teach: How can changes work together?

1. Start with y = x^2.
2. Example: y = -2(x - 1)^2 + 3.
3. Stretch vertically by 2.
4. Reflect across the x-axis.
5. Then shift right 1 and up 3.
6. The new vertex is (1, 3).
7. The parabola opens downward.
8. Its maximum output is 3.
9. Try: Why is 3 a maximum here?

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16. Factor the Inside First

Teach: Why factor before shifting?

1. The raw constant can mislead you.
2. Example: g(x) = |2x - 6|.
3. Factor: g(x) = |2(x - 3)|.
4. The horizontal shift is right 3.
5. It is not a shift right 6.
6. There is also a width factor of 1/2.
7. The vertex occurs when 2x - 6 = 0.
8. So the vertex is (3, 0).
9. Try: Find the vertex of |2x - 4|.

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17. Read Slope-Intercept Form

Teach: What does y = mx + b tell us?

1. m is the slope, or rate of change.
2. b is the y-intercept's y-value.
3. Example: y = 25x + 40.
4. A gym charges $40 to join.
5. Each month adds another $25.
6. Let x be months and y be dollars.
7. At x = 0, the total is $40.
8. At x = 2, y = 25(2) + 40 = 90.
9. Try: Explain the 25 in words.

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18. Write a Line from a Point

Teach: What is point-slope form?

1. Use y - y1 = m(x - x1).
2. You need a slope and one point.
3. Example: m = 2 through (1, 3).
4. Substitute: y - 3 = 2(x - 1).
5. Expand: y - 3 = 2x - 2.
6. Add 3: y = 2x + 1.
7. Check the point by using x = 1.
8. Then y = 2(1) + 1 = 3.
9. Try: Write a line, m = 3, at (0, 2).

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19. Explain an Intersection

Teach: What solves a system?

1. One point must satisfy both rules.
2. Graph both lines on the same axes.
3. Their intersection is the solution.
4. Example: y = 2x - 1.
5. Also graph y = -x + 5.
6. They meet at (2, 3).
7. First check: 3 = 2(2) - 1.
8. Second check: 3 = -2 + 5.
9. Try: Why must both checks work?

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20. Count System Solutions

Teach: Can two lines fail to meet?

1. Yes: distinct parallel lines do not.
2. They give a system with no solution.
3. Different slopes give one crossing.
4. That means exactly one solution.
5. Identical lines share every point.
6. They give infinitely many solutions.
7. Compare y = 2x + 1 and y = 2x - 3.
8. Same slope, different intercepts.
9. Try: How many solutions? Explain.

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21. Solve by Substitution

Teach: What does substitution do?

1. Replace a variable with its equal.
2. Example: y = 2x and x + y = 6.
3. Replace y in the second equation.
4. Then x + 2x = 6.
5. Combine: 3x = 6, so x = 2.
6. Use y = 2x to get y = 4.
7. The solution is (2, 4).
8. Check: 4 = 2(2) and 2 + 4 = 6.
9. Try: Explain why we replaced y.

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22. Solve by Elimination

Teach: What does elimination do?

1. It cancels one variable's terms.
2. Example: x + y = 5.
3. Second equation: x - y = 1.
4. Add the equations: 2x = 6.
5. The +y and -y cancel each other.
6. So x = 3.
7. Use 3 + y = 5 to get y = 2.
8. Check: 3 + 2 = 5 and 3 - 2 = 1.
9. Try: When would subtraction help?

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23. Draw and Shade a Boundary

Teach: How do we graph y > x + 1?

1. First draw the boundary y = x + 1.
2. Use dashes because > excludes =.
3. Shade above: y must be larger.
4. For >=, use a solid boundary.
5. Test a point away from the line.
6. Try (0, 3): 3 > 1 is true.
7. So its side must be shaded.
8. Flip the sign if dividing by -1.
9. Try: Graph y <= x + 1 instead.

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24. Keep Only the Overlap

Teach: What solves two inequalities?

1. A point must satisfy both at once.
2. Example: y >= x and y <= 4.
3. Shade above or on y = x.
4. Shade below or on y = 4.
5. Keep only the shared region.
6. (1, 3) works: 3 >= 1 and 3 <= 4.
7. (5, 2) fails because 2 < 5.
8. No overlap means no solution.
9. Try: Does (2, 4) satisfy both?

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25. Goal Versus Limits

Teach: What is linear programming?

1. It finds a best value within limits.
2. The objective states your goal.
3. Constraints state the limits.
4. Example goal: maximize P = 3x + 2y.
5. Require x >= 0 and y >= 0.
6. Also require x + y <= 8.
7. The feasible region meets all limits.
8. Maximize means find the largest P.
9. Try: What does minimize mean?

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26. Test the Corners

Teach: How can corners find the best?

1. Use a closed, bounded polygon.
2. For a linear goal, test each vertex.
3. Example: x >= 0, y >= 0, x + y <= 4.
4. Vertices: (0, 0), (4, 0), (0, 4).
5. Maximize P = 3x + 2y.
6. The values are 0, 12, and 8.
7. The maximum is 12 at (4, 0).
8. Check that point meets every limit.
9. Try: Find the minimum P here.

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