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Step-by-step solution for: SOLUTIONS-CHAPTER-2-Holt-Algebra-2-2007 key - CHAPTER Solutions ...
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Step-by-step solution for: SOLUTIONS-CHAPTER-2-Holt-Algebra-2-2007 key - CHAPTER Solutions ...
Here are the solutions to the problems on the worksheet.
1. True or False
Answer: True
Explanation: By definition, a "zero" of a function is an $x$-value that makes $f(x) = 0$. On a graph, points where the $y$-value is 0 lie on the x-axis. Therefore, if $k$ is a zero, the point $(k, 0)$ is on the graph, which means it is an x-intercept.
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Determine the lowest-degree polynomial (Problems 2–4)
*Note: We determine the degree by counting the number of "turns" (bumps) in the graph and how the ends behave.*
2. The graph has 3 turns. It starts down and ends up (odd degree behavior).
* Minimum degree = Turns + 1 = $3 + 1 = 4$? No, wait. An odd degree graph must have an even number of turns usually? Let's look closer.
* Actually, simpler rule: Count the bumps. There are 3 local extrema (peaks/valleys). The minimum degree is $n+1$ where $n$ is the number of turning points. So, degree 4?
* Let's check end behavior: Left side goes Down, Right side goes Up. This is characteristic of an Odd degree (like $x^3, x^5$).
* An odd degree polynomial with 3 turns must be at least degree 4? No, $x^3$ has 2 turns max? No, $x^3$ has 1 turn (inflection) or 2 extrema? A cubic ($x^3$) can have 2 turning points. A quartic ($x^4$) can have 3. But quartics have same-direction ends (both up or both down). This graph has opposite ends.
* Correction: Opposite ends = Odd Degree. Same ends = Even Degree.
* Graph 2: Ends are opposite (Down/Up). Must be Odd. It has 3 turns. The lowest odd degree with 3 turns is Degree 5? Or is it Degree 3 with extra wiggles?
* Let's re-evaluate standard curriculum rules.
* Rule: Max turns = $n - 1$. So if there are 3 turns, $n - 1 \ge 3 \rightarrow n \ge 4$.
* However, if $n=4$ (even), ends must match. They don't.
* If $n=5$ (odd), ends oppose. They do. And 5 allows for up to 4 turns. This graph has 3. So Degree 5 is possible.
* *Wait*, let's look at Graph 2 again. It crosses the axis 3 times. It has a local min, local max, local min. That is 3 turning points.
* Actually, looking closely at Graph 2: It comes from bottom left, goes up, turns down, turns up, turns down... wait.
* Let's trace Graph 2 carefully: Starts low (left), goes up to a peak, down to a valley, up to a peak, down to a valley, then up? No.
* Let's look at the image provided.
* Graph 2: Starts Bottom-Left, goes Up, turns, goes Down, turns, goes Up, turns, goes Down... actually it looks like it ends Top-Right.
* Path: Up -> Turn -> Down -> Turn -> Up -> End.
* This has 2 turns.
* Ends: Down / Up (Opposite).
* Lowest degree with 2 turns and opposite ends is 3 (Cubic).
* Graph 3: Starts Top-Left, goes Down, turns, goes Up, turns, goes Down, turns, goes Up.
* Ends: Up / Up (Same). This means Even degree.
* Turns: 3 turns.
* Lowest even degree with 3 turns is 4 (Quartic).
* Graph 4: Starts Bottom-Left, goes Up, flattens/turns slightly?, goes Up.
* It looks like an inflection point. It crosses the x-axis once.
* Ends: Down / Up (Opposite). Odd degree.
* Turns: It looks like it has 2 turns (a small wiggle). Or maybe just 1 inflection?
* Let's look closer. It goes up, bends right, bends left, goes up. That is 2 turns.
* Lowest odd degree with 2 turns is 3.
* *Self-Correction on Graph 2*: Looking really closely at crop 2. It crosses x-axis at -2, 0, 2 roughly. It has a peak between -2 and 0, and a valley between 0 and 2. That is 2 turns. Ends are opposite. Degree = 3.
* *Self-Correction on Graph 3*: Crosses x-axis 4 times. Has 3 peaks/valleys. Ends are same (Up/Up). Degree = 4.
* *Self-Correction on Graph 4*: Crosses x-axis once. Has a "wiggle" (slope changes from steep to flat to steep). This implies 2 turns. Ends opposite. Degree = 3.
Answers:
2. Degree 3
3. Degree 4
4. Degree 3
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Estimate coordinates of turning points (Problems 5–7)
5.
* Local Maximum (Peak): The highest point in the middle bump is at $(0, 2)$.
* Local Minimum (Valley): The lowest point on the right is at $(1, -1)$.
* *(Note: There is another valley on the far left around $x=-1.5, y=-5$, but usually these problems ask for the visible local ones near the origin or all of them. Let's list the clear ones).*
* Looking at the grid:
* Peak at $x=0, y=2$.
* Valley at $x=1, y=-1$.
* Valley at $x \approx -1.5, y \approx -5$.
6.
* Local Minimum (Valley): The bottom of the U-shape is at $(1, -2)$.
7.
* Local Maximum (Peak): The small hill is at $(-1, 1)$.
* Local Minimum (Valley): The deep dip is at $(1, -3)$.
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Match the graph with its function (Problems 8–10)
8. $f(x) = 2x^4 - 3x^2 - 2$
* End Behavior: Positive leading coefficient ($2x^4$) and even power. Both ends go UP.
* Y-intercept: Plug in $x=0 \rightarrow y = -2$.
* Graph: Look for a "W" shape opening up, crossing y-axis at -2.
* Match: Graph C opens up and crosses y at -2. (Graph B crosses at -10, Graph A opens down).
9. $f(x) = 2x^6 - 6x^4 + 4x^2 - 210$
* End Behavior: Positive leading coefficient ($2x^6$) and even power. Both ends go UP.
* Y-intercept: Plug in $x=0 \rightarrow y = -210$. This is very low.
* Graph: Look for a "W" or "M" shape opening up, but the middle part is way down below the visible grid or just barely visible.
* Match: Graph B. It opens up, but the vertex is very deep (the scale shows -10, and the graph goes way past it).
10. $f(x) = -2x^4 + 3x^2 - 2$
* End Behavior: Negative leading coefficient ($-2x^4$) and even power. Both ends go DOWN.
* Y-intercept: Plug in $x=0 \rightarrow y = -2$.
* Graph: Look for an "M" shape opening down.
* Match: Graph A.
Answers:
8. C
9. B
10. A
---
Determine the x-intercepts (Problems 11–16)
*To find x-intercepts, set each factor equal to zero and solve for x.*
11. $g(x) = (x + 4)(x - 1)$
* $x + 4 = 0 \rightarrow x = -4$
* $x - 1 = 0 \rightarrow x = 1$
* Intercepts: $-4, 1$
12. $h(x) = (x - 2)(x - 3)$
* $x - 2 = 0 \rightarrow x = 2$
* $x - 3 = 0 \rightarrow x = 3$
* Intercepts: $2, 3$
13. $f(x) = x(x + 4)(x - 5)$
* $x = 0$
* $x + 4 = 0 \rightarrow x = -4$
* $x - 5 = 0 \rightarrow x = 5$
* Intercepts: $0, -4, 5$
14. $f(x) = (x + 3)(x + 1)(x - 8)$
* $x + 3 = 0 \rightarrow x = -3$
* $x + 1 = 0 \rightarrow x = -1$
* $x - 8 = 0 \rightarrow x = 8$
* Intercepts: $-3, -1, 8$
15. $g(x) = (x + 6)^2$
* $x + 6 = 0 \rightarrow x = -6$
* Intercept: $-6$
16. $h(x) = (x - 1)(x - 7)^2$
* $x - 1 = 0 \rightarrow x = 1$
* $x - 7 = 0 \rightarrow x = 7$
* Intercepts: $1, 7$
──────────────────────────────────────
Final Answer:
1. True
Lowest-Degree Polynomial:
2. 3
3. 4
4. 3
Turning Points (Estimates):
5. Local Max: $(0, 2)$; Local Min: $(1, -1)$ [and approx $(-1.5, -5)$]
6. Local Min: $(1, -2)$
7. Local Max: $(-1, 1)$; Local Min: $(1, -3)$
Match Graph to Function:
8. C
9. B
10. A
X-Intercepts:
11. $x = -4, 1$
12. $x = 2, 3$
13. $x = 0, -4, 5$
14. $x = -3, -1, 8$
15. $x = -6$
16. $x = 1, 7$
1. True or False
Answer: True
Explanation: By definition, a "zero" of a function is an $x$-value that makes $f(x) = 0$. On a graph, points where the $y$-value is 0 lie on the x-axis. Therefore, if $k$ is a zero, the point $(k, 0)$ is on the graph, which means it is an x-intercept.
---
Determine the lowest-degree polynomial (Problems 2–4)
*Note: We determine the degree by counting the number of "turns" (bumps) in the graph and how the ends behave.*
2. The graph has 3 turns. It starts down and ends up (odd degree behavior).
* Minimum degree = Turns + 1 = $3 + 1 = 4$? No, wait. An odd degree graph must have an even number of turns usually? Let's look closer.
* Actually, simpler rule: Count the bumps. There are 3 local extrema (peaks/valleys). The minimum degree is $n+1$ where $n$ is the number of turning points. So, degree 4?
* Let's check end behavior: Left side goes Down, Right side goes Up. This is characteristic of an Odd degree (like $x^3, x^5$).
* An odd degree polynomial with 3 turns must be at least degree 4? No, $x^3$ has 2 turns max? No, $x^3$ has 1 turn (inflection) or 2 extrema? A cubic ($x^3$) can have 2 turning points. A quartic ($x^4$) can have 3. But quartics have same-direction ends (both up or both down). This graph has opposite ends.
* Correction: Opposite ends = Odd Degree. Same ends = Even Degree.
* Graph 2: Ends are opposite (Down/Up). Must be Odd. It has 3 turns. The lowest odd degree with 3 turns is Degree 5? Or is it Degree 3 with extra wiggles?
* Let's re-evaluate standard curriculum rules.
* Rule: Max turns = $n - 1$. So if there are 3 turns, $n - 1 \ge 3 \rightarrow n \ge 4$.
* However, if $n=4$ (even), ends must match. They don't.
* If $n=5$ (odd), ends oppose. They do. And 5 allows for up to 4 turns. This graph has 3. So Degree 5 is possible.
* *Wait*, let's look at Graph 2 again. It crosses the axis 3 times. It has a local min, local max, local min. That is 3 turning points.
* Actually, looking closely at Graph 2: It comes from bottom left, goes up, turns down, turns up, turns down... wait.
* Let's trace Graph 2 carefully: Starts low (left), goes up to a peak, down to a valley, up to a peak, down to a valley, then up? No.
* Let's look at the image provided.
* Graph 2: Starts Bottom-Left, goes Up, turns, goes Down, turns, goes Up, turns, goes Down... actually it looks like it ends Top-Right.
* Path: Up -> Turn -> Down -> Turn -> Up -> End.
* This has 2 turns.
* Ends: Down / Up (Opposite).
* Lowest degree with 2 turns and opposite ends is 3 (Cubic).
* Graph 3: Starts Top-Left, goes Down, turns, goes Up, turns, goes Down, turns, goes Up.
* Ends: Up / Up (Same). This means Even degree.
* Turns: 3 turns.
* Lowest even degree with 3 turns is 4 (Quartic).
* Graph 4: Starts Bottom-Left, goes Up, flattens/turns slightly?, goes Up.
* It looks like an inflection point. It crosses the x-axis once.
* Ends: Down / Up (Opposite). Odd degree.
* Turns: It looks like it has 2 turns (a small wiggle). Or maybe just 1 inflection?
* Let's look closer. It goes up, bends right, bends left, goes up. That is 2 turns.
* Lowest odd degree with 2 turns is 3.
* *Self-Correction on Graph 2*: Looking really closely at crop 2. It crosses x-axis at -2, 0, 2 roughly. It has a peak between -2 and 0, and a valley between 0 and 2. That is 2 turns. Ends are opposite. Degree = 3.
* *Self-Correction on Graph 3*: Crosses x-axis 4 times. Has 3 peaks/valleys. Ends are same (Up/Up). Degree = 4.
* *Self-Correction on Graph 4*: Crosses x-axis once. Has a "wiggle" (slope changes from steep to flat to steep). This implies 2 turns. Ends opposite. Degree = 3.
Answers:
2. Degree 3
3. Degree 4
4. Degree 3
---
Estimate coordinates of turning points (Problems 5–7)
5.
* Local Maximum (Peak): The highest point in the middle bump is at $(0, 2)$.
* Local Minimum (Valley): The lowest point on the right is at $(1, -1)$.
* *(Note: There is another valley on the far left around $x=-1.5, y=-5$, but usually these problems ask for the visible local ones near the origin or all of them. Let's list the clear ones).*
* Looking at the grid:
* Peak at $x=0, y=2$.
* Valley at $x=1, y=-1$.
* Valley at $x \approx -1.5, y \approx -5$.
6.
* Local Minimum (Valley): The bottom of the U-shape is at $(1, -2)$.
7.
* Local Maximum (Peak): The small hill is at $(-1, 1)$.
* Local Minimum (Valley): The deep dip is at $(1, -3)$.
---
Match the graph with its function (Problems 8–10)
8. $f(x) = 2x^4 - 3x^2 - 2$
* End Behavior: Positive leading coefficient ($2x^4$) and even power. Both ends go UP.
* Y-intercept: Plug in $x=0 \rightarrow y = -2$.
* Graph: Look for a "W" shape opening up, crossing y-axis at -2.
* Match: Graph C opens up and crosses y at -2. (Graph B crosses at -10, Graph A opens down).
9. $f(x) = 2x^6 - 6x^4 + 4x^2 - 210$
* End Behavior: Positive leading coefficient ($2x^6$) and even power. Both ends go UP.
* Y-intercept: Plug in $x=0 \rightarrow y = -210$. This is very low.
* Graph: Look for a "W" or "M" shape opening up, but the middle part is way down below the visible grid or just barely visible.
* Match: Graph B. It opens up, but the vertex is very deep (the scale shows -10, and the graph goes way past it).
10. $f(x) = -2x^4 + 3x^2 - 2$
* End Behavior: Negative leading coefficient ($-2x^4$) and even power. Both ends go DOWN.
* Y-intercept: Plug in $x=0 \rightarrow y = -2$.
* Graph: Look for an "M" shape opening down.
* Match: Graph A.
Answers:
8. C
9. B
10. A
---
Determine the x-intercepts (Problems 11–16)
*To find x-intercepts, set each factor equal to zero and solve for x.*
11. $g(x) = (x + 4)(x - 1)$
* $x + 4 = 0 \rightarrow x = -4$
* $x - 1 = 0 \rightarrow x = 1$
* Intercepts: $-4, 1$
12. $h(x) = (x - 2)(x - 3)$
* $x - 2 = 0 \rightarrow x = 2$
* $x - 3 = 0 \rightarrow x = 3$
* Intercepts: $2, 3$
13. $f(x) = x(x + 4)(x - 5)$
* $x = 0$
* $x + 4 = 0 \rightarrow x = -4$
* $x - 5 = 0 \rightarrow x = 5$
* Intercepts: $0, -4, 5$
14. $f(x) = (x + 3)(x + 1)(x - 8)$
* $x + 3 = 0 \rightarrow x = -3$
* $x + 1 = 0 \rightarrow x = -1$
* $x - 8 = 0 \rightarrow x = 8$
* Intercepts: $-3, -1, 8$
15. $g(x) = (x + 6)^2$
* $x + 6 = 0 \rightarrow x = -6$
* Intercept: $-6$
16. $h(x) = (x - 1)(x - 7)^2$
* $x - 1 = 0 \rightarrow x = 1$
* $x - 7 = 0 \rightarrow x = 7$
* Intercepts: $1, 7$
──────────────────────────────────────
Final Answer:
1. True
Lowest-Degree Polynomial:
2. 3
3. 4
4. 3
Turning Points (Estimates):
5. Local Max: $(0, 2)$; Local Min: $(1, -1)$ [and approx $(-1.5, -5)$]
6. Local Min: $(1, -2)$
7. Local Max: $(-1, 1)$; Local Min: $(1, -3)$
Match Graph to Function:
8. C
9. B
10. A
X-Intercepts:
11. $x = -4, 1$
12. $x = 2, 3$
13. $x = 0, -4, 5$
14. $x = -3, -1, 8$
15. $x = -6$
16. $x = 1, 7$
Parent Tip: Review the logic above to help your child master the concept of holt algebra 2 worksheet.
