Key features of graphs and tables

1. The y-intercept is the point where the graph crosses the y-axis. Every point on the y-axis has x = 0.
2. Follow the y-axis (the vertical line through 0) until it meets the curve.
3. The curve crosses it at y = -3.

The y-intercept is (0,\ -3).

Check: The function graphed is y = (x - 1)^2 - 4. At x = 0: (0 - 1)^2 - 4 = 1 - 4 = -3 ✓

Tip: A function has at most one y-intercept but may have several x-intercepts. If you see two answers, you're probably looking at the x-axis.

1. The highest point of the curve is its maximum.
2. The top of the curve is directly above t = 2 seconds.
3. Read across from the top to the vertical (height) axis: it lines up with 64.

The maximum height is 64 feet (reached at 2 seconds).

Check: The curve is symmetric: it leaves the ground at t = 0 and lands at t = 4, so the top is halfway, at t = 2. The formula h = -16t^2 + 64t gives h(2) = -64 + 128 = 64 ✓

Tip: "Maximum height" asks for a y-value; "when is it highest" asks for an x-value. Underline which one the question wants.

1. Read from left to right:
   • From x = -5 to x = -2 the graph climbs from -3 to 3 (increasing).
   • From x = -2 to x = 1 it falls from 3 to -3 (decreasing).
   • From x = 1 to x = 4 it climbs again (increasing).
2. Intervals are written with x-values, so the decreasing interval is -2 < x < 1.

Answer: -2 < x < 1.

Check: Pick x = -1 and x = 0 inside the interval: g(-1) = 1 and g(0) = -1. As x grows, g shrinks ✓

Tip: Increasing/decreasing intervals describe where on the x-axis the change happens. The heights -3 and 3 describe how far it falls, not where.

1. A period is the horizontal length of one complete cycle.
2. The seat is at its lowest (5 ft) at t = 0, rises to 85 ft at t = 4, and is back at its lowest at t = 8.
3. Peak to peak gives the same result: peaks at t = 4 and t = 12, so 12 - 4 = 8.

The period is 8 minutes: the wheel makes one full turn every 8 minutes.

Check: Low points at t = 0, 8, 16, 24 — each 8 minutes apart ✓

Tip: The period is measured along the horizontal (time) axis. The distance between the max and min heights (85 - 5 = 80 ft) is a vertical measurement and has nothing to do with the period.

1. A function is decreasing where its graph goes down as you move left to right.
2. From t = 0 to t = 2 the curve goes up (increasing).
3. From t = 2 to t = 4 the curve goes down (decreasing), until the ball hits the ground.
4. Intervals are written with x-values (here, times).

The height is decreasing for 2 < t < 4.

Check: Pick t = 3: h = -16(9) + 64(3) = -144 + 192 = 48, which is less than the 64 ft at t = 2 — still going down ✓

Tip: Increasing/decreasing intervals are always described with x-values, read left to right, and they change at the maximum or minimum.

1. Minimum: the smallest output is -4, at x = 0. The values go down to -4 and then back up. So the minimum value -4 occurs at x = 0.
2. Symmetry: f(-1) = f(1) = -3, f(-2) = f(2) = 0, f(-3) = f(3) = 5 — mirror images about x = 0.
3. x-intercepts: f(x) = 0 at x = -2 and x = 2.
4. y-intercept: f(0) = -4.

The true statement is "The minimum value of f is -4, at x = 0."

Check: The table matches f(x) = x^2 - 4: (-3)^2 - 4 = 5, 0^2 - 4 = -4 ✓

Tip: In a table, the y-intercept is the output in the x = 0 column, and the x-intercepts are the columns where the output is 0.

Each segment covers 4 hours, so compare the temperature changes:

1. 0–4: 42 - 48 = -6 (falling)
2. 4–8: 50 - 42 = +8
3. 8–12: 66 - 50 = +16
4. 12–16: 70 - 66 = +4
5. 16–20: 58 - 70 = -12 (falling)
6. 20–24: 50 - 58 = -8 (falling)

The biggest increase is +16°F, from 8 to 12 hours after midnight (8 a.m. to noon). That is \frac{16}{4} = 4 degrees per hour.

Check: On the graph the 8–12 segment is the steepest one going up ✓

Tip: "Fastest increase" = steepest rising segment = greatest positive slope. The highest point (70°F) is not where the change is fastest.

1. A graph of the profit crosses the horizontal axis (profit = 0) wherever the values change sign.
2. Scan the row from left to right:
   • Month 1 to 2: -4 \to -1 (still a loss).
   • Month 2 to 3: -1 \to 2 — negative to positive. The profit passes through 0 here.
3. The later sign change, month 6 to 7 (1 \to -2), goes from a gain back to a loss.

The profit first changed from a loss to a gain between months 2 and 3.

Check: -1 < 0 < 2, so 0 lies between the month-2 and month-3 values ✓

Tip: In a table, zeros (intercepts) hide between rows when the sign flips. The largest value (here 3, the maximum) is a different feature.

Find the change in height for each 2-week period:

1. Weeks 0–2: 5 - 2 = 3 inches
2. Weeks 2–4: 12 - 5 = 7 inches
3. Weeks 4–6: 15 - 12 = 3 inches
4. Weeks 6–8: 16 - 15 = 1 inch

The smallest change is 1 inch, so the plant grew slowest during weeks 6–8 (a rate of \frac{1}{2} inch per week).

Check: On the graph, the segment from (6, 15) to (8, 16) is the flattest ✓

Tip: Rate of change is about steepness. A segment can be high on the graph and still be nearly flat. Weeks 0–2 are the lowest heights, but the plant grew 3 inches then.

1. "As x increases without bound" means: follow the graph far to the right.
2. Past the maximum at (-1, 9), the curve keeps falling. It crosses the x-axis at (2, 0) and keeps going down, faster and faster.
3. Nothing stops it: the right end points downward forever.

So f(x) decreases without bound (it goes toward -\infty).

Check: f(x) = -x^2 - 2x + 8: f(5) = -25 - 10 + 8 = -27, f(10) = -100 - 20 + 8 = -112 — more and more negative ✓

Tip: A parabola that opens down has both ends pointing down; one that opens up has both ends pointing up. Crossing the x-axis does not mean the graph stops there.

Test the graphs feature by feature.

1. Maximum at (0, 4): Graph B has a minimum at (0, -4) — out. Graph C peaks at (2, 4) — out. A and D both peak at (0, 4).
2. x-intercepts -2 and 2: Graph A crosses at (-2, 0) and (2, 0) ✓. Graph D crosses at (-4, 0) and (4, 0) — out.
3. Increasing then decreasing: Graph A rises up to x = 0 and falls after ✓.

The graph is Graph A.

Check: Graph A matches f(x) = 4 - x^2: f(0) = 4 ✓, f(\pm 2) = 4 - 4 = 0 ✓

Tip: With "which graph" questions, use the most specific feature first (a named point like the maximum), then eliminate. Make sure the final choice satisfies every feature.

1. The horizontal axis is the ticket price; the vertical axis is the daily profit.
2. At an x-intercept the profit is 0: the theater breaks even (no gain, no loss).
3. The x-values are 5 and 35, so these are the ticket prices $5 and $35.
4. Between these prices the profit is positive; below $5 or above $35 the theater loses money.

Answer: At ticket prices of $5 and $35, the theater's profit is $0 (it breaks even).

Check: The model is P = -4(x - 5)(x - 35); at x = 5 or x = 35 one factor is 0, so P = 0 ✓

Tip: In context, always say what both numbers in an intercept mean, with units: (5, 0) = "price $5, profit $0". The best price is a different feature — the maximum (20, 900).

1. Maximum of f: the values rise to 9 at x = 1 and fall again symmetrically (8, 8 and 5, 5). So the maximum of f is 9.
2. Maximum of g: a = -1, b = 6. x = -\frac{6}{2(-1)} = 3, \qquad g(3) = -(3)^2 + 6(3) + 1 = -9 + 18 + 1 = 10 So the maximum of g is 10.
3. Compare: 10 > 9.

Answer: The maximum of g (10) is greater than the maximum of f (9).

Check: g(2) = -4 + 12 + 1 = 9 and g(4) = -16 + 24 + 1 = 9, both below 10 ✓

Tip: Compare the same feature on both sides: maximum value against maximum value. The 1 in g(x) is its y-intercept, and 3 is where its maximum occurs — neither is the maximum value.

1. Start: 40 gallons at hour 0 → the graph begins at (0,\ 40). Graph C starts at 0 — out.
2. First 2 hours, no change: a horizontal piece from (0, 40) to (2, 40). Graph B drops immediately — out.
3. Steady draining: a straight line down (constant negative slope).
4. Empty at hour 6: the line ends at (6,\ 0). Graph D reaches 0 at hour 4 — out.

The correct graph is Graph A.

Check: Graph A's draining rate: \frac{0 - 40}{6 - 2} = -10 gallons per hour — steady ✓

Tip: A horizontal segment means "no change"; a steady change is a straight slanted segment. Check that the pieces appear in the same order as the story.

1. The coefficient of x^2 is -1 (negative), so the parabola opens downward and has a maximum, not a minimum.
2. The turning point is at x = -\frac{b}{2a} = -\frac{6}{2(-1)} = 3
3. The maximum value: f(3) = -(3)^2 + 6(3) - 5 = -9 + 18 - 5 = 4

The true statement is "f has a maximum value of 4 at x = 3."

Check: Symmetry test: f(2) = -4 + 12 - 5 = 3 and f(4) = -16 + 24 - 5 = 3 — equal values on both sides of x = 3, both less than 4 ✓

Tip: Not on the formula sheet: the turning point of y = ax^2 + bx + c is at x = -\frac{b}{2a}. If a < 0 it's a maximum; if a > 0 it's a minimum. You can also make a quick table of values with the calculator.

1. The vertical axis is speed, not distance.
2. From t = 2 to t = 6 the graph is horizontal at 30: the speed stays at 30 mi/h.
3. A constant speed of 30 mi/h means the van is moving steadily, not stopped. (It is stopped only where the speed is 0, at t = 0 and t = 7.)
4. For comparison: from t = 0 to t = 2 the speed increases (speeding up); from t = 6 to t = 7 it decreases (slowing to a stop).

Answer: The van travels at a constant speed of 30 miles per hour.

Check: In 4 minutes at 30 mi/h, the van covers 30 \times \frac{4}{60} = 2 miles — so it is definitely moving ✓

Tip: A flat piece means "not changing" for whatever the vertical axis measures. Flat on a distance graph = stopped; flat on a speed graph = steady speed.

From the graph of f: maximum (-1,\ 9), y-intercept 8, x-intercepts -4 and 2.

For g(x) = -x^2 + 2x + 15:

1. Vertex: x = -\frac{2}{2(-1)} = 1, and g(1) = -1 + 2 + 15 = 16. Maximum (1,\ 16).
2. y-intercept: g(0) = 15.
3. x-intercepts: -x^2 + 2x + 15 = 0 \Rightarrow x^2 - 2x - 15 = 0 \Rightarrow (x - 5)(x + 3) = 0, so x = 5 or x = -3.

Compare:

• Maximum: 16 > 9, so g has the greater maximum ✓
• y-intercept: 15 > 8, so f does not have the greater y-intercept.
• g crosses at -3 and 5, not 3 and -5.
• The maxima occur at x = -1 (for f) and x = 1 (for g) — different.

Answer: g has a greater maximum value than f.

Check: g(5) = -25 + 10 + 15 = 0 ✓ and g(-3) = -9 - 6 + 15 = 0 ✓

Tip: From factors (x - 5)(x + 3), the zeros have the opposite signs: x = 5 and x = -3.

1. Right end ("as x increases without bound"): after the last x-intercept at (3, 0), the curve heads down toward the lower right. So f(x) decreases without bound.
2. Left end ("as x decreases without bound"): before (-2, 0), the curve comes down from the upper left. Traced to the left, it rises higher and higher. So f(x) increases without bound.
3. The bumps in the middle (the turning points) do not affect end behavior.

Answer: As x increases without bound, f(x) decreases without bound; as x decreases without bound, f(x) increases without bound.

Check: The curve has zeros -2, 1, 3 and y-intercept -3, which fits f(x) = -\tfrac{1}{2}(x + 2)(x - 1)(x - 3). Its leading term -\tfrac{1}{2}x^3 is negative for large positive x and positive for large negative x ✓

Tip: Describe each end by tracing outward from the middle of the graph. "Comes down from the upper left" means that, going left, the graph goes up.

1. The point (0,\ -700) is the y-intercept: ticket price x = 0.
2. Price $0 means the tickets are free.
3. Profit -700 means a loss of $700 per day (costs are not covered by any ticket sales).
4. So: if the theater gave away its tickets, it would lose $700 each day.

Answer: If tickets were free, the theater would lose $700 per day.

Check: Model P = -4(x - 5)(x - 35): P(0) = -4(-5)(-35) = -4(175) = -700 ✓

Tip: A negative output in a money context means a loss. Read the coordinates in order (input, output) and attach the units from each axis.

1. Ball A (graph): the top of the curve is at (2,\ 64), so its maximum height is 64 ft.
2. Ball B (equation): find when h = 0: -16t^2 + 80t = 0 \Rightarrow -16t(t - 5) = 0 \Rightarrow t = 0 \text{ or } t = 5
3. By symmetry, the maximum is halfway: t = \frac{0 + 5}{2} = 2.5 s. h(2.5) = -16(2.5)^2 + 80(2.5) = -16(6.25) + 200 = -100 + 200 = 100
4. Compare: 100 - 64 = 36.

Ball B goes higher, by 36 feet.

Check: t = -\frac{b}{2a} = -\frac{80}{2(-16)} = 2.5 ✓, and h(2) = -64 + 160 = 96 < 100, h(3) = -144 + 240 = 96 < 100 — symmetric around 2.5 ✓

Tip: To compare a feature across two representations, compute the same feature (max height) for each one and then compare the numbers.

1. a = -16, b = 96, c = 4. Since a < 0, the graph opens down and has a maximum.
2. Time of maximum: t = -\frac{b}{2a} = -\frac{96}{2(-16)} = \frac{96}{32} = 3 \text{ seconds}
3. Maximum height: h = -16(3)^2 + 96(3) + 4 = -16(9) + 288 + 4 = -144 + 288 + 4 = 148

Answer: The rocket reaches a maximum height of 148 feet after 3 seconds.

Check: One second earlier and later give the same, smaller height (symmetry): h(2) = -64 + 192 + 4 = 132 and h(4) = -256 + 384 + 4 = 132 ✓

Tip: Square first, then multiply by -16: -16(3)^2 = -16 \cdot 9 = -144. And keep the "+4" — it is the platform height, part of every height.

Check each graph against all three features:

Only Graph A has every feature.

Check: Graph A matches f(x) = \tfrac{1}{2}(x - 2)^3: f(2) = 0 ✓, f(0) = \tfrac{1}{2}(-8) = -4 ✓, and it rises from lower left to upper right ✓

Tip: A table like the one above is a fast, reliable way to handle "which graph has all these features" — the GED often builds each wrong choice to match all but one feature.