Math / Algebra

Question2. Sketch the following graphs on separate diagrams. (a) $y=(x+a)^{3}, a>0$ (b) $y=x^{3}-b, b>0$

Studdy Solution

STEP 1

1. We are dealing with cubic functions, which have the general form $y = x^3$ .
2. The transformations involve horizontal and vertical shifts.
3. The parameter $a$ in part (a) represents a horizontal shift.
4. The parameter $b$ in part (b) represents a vertical shift.

STEP 2

1. Analyze and sketch the graph for $y = (x+a)^3$ .
2. Analyze and sketch the graph for $y = x^3 - b$ .

STEP 3

The base function is $y = x^3$ , which is a cubic function with an inflection point at the origin (0,0).

STEP 4

For $y = (x+a)^3$ , the graph of $y = x^3$ is shifted horizontally to the left by $a$ units because $a > 0$ .

STEP 5

Sketch the graph of $y = (x+a)^3$ : - Start with the graph of $y = x^3$ . - Shift every point on this graph $a$ units to the left. - The inflection point moves from (0,0) to (-a,0).

STEP 6

The base function is again $y = x^3$ .

STEP 7

For $y = x^3 - b$ , the graph of $y = x^3$ is shifted vertically downward by $b$ units because $b > 0$ .

STEP 8

Sketch the graph of $y = x^3 - b$ : - Start with the graph of $y = x^3$ . - Shift every point on this graph $b$ units downward. - The inflection point moves from (0,0) to (0,-b).
The graphs for each function have been sketched with the described transformations.

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Question2. Sketch the following graphs on separate diagrams. (a) $y=(x+a)^{3}, a>0$ (b) $y=x^{3}-b, b>0$

Studdy Solution

STEP 1

1. We are dealing with cubic functions, which have the general form $y = x^3$ .
2. The transformations involve horizontal and vertical shifts.
3. The parameter $a$ in part (a) represents a horizontal shift.
4. The parameter $b$ in part (b) represents a vertical shift.

STEP 2

1. Analyze and sketch the graph for $y = (x+a)^3$ .
2. Analyze and sketch the graph for $y = x^3 - b$ .

STEP 3

The base function is $y = x^3$ , which is a cubic function with an inflection point at the origin (0,0).

STEP 4

For $y = (x+a)^3$ , the graph of $y = x^3$ is shifted horizontally to the left by $a$ units because $a > 0$ .

STEP 5

Sketch the graph of $y = (x+a)^3$ : - Start with the graph of $y = x^3$ . - Shift every point on this graph $a$ units to the left. - The inflection point moves from (0,0) to (-a,0).

STEP 6

The base function is again $y = x^3$ .

STEP 7

For $y = x^3 - b$ , the graph of $y = x^3$ is shifted vertically downward by $b$ units because $b > 0$ .

STEP 8

Sketch the graph of $y = x^3 - b$ : - Start with the graph of $y = x^3$ . - Shift every point on this graph $b$ units downward. - The inflection point moves from (0,0) to (0,-b).
The graphs for each function have been sketched with the described transformations.

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Question2. Sketch the following graphs on separate diagrams. (a) y=(x+a)3,a>0y=(x+a)^{3}, a>0y=(x+a)3,a>0 (b) y=x3−b,b>0y=x^{3}-b, b>0y=x3−b,b>0

Studdy Solution

STEP 1

1. We are dealing with cubic functions, which have the general form y=x3 y = x^3 y=x3.2. The transformations involve horizontal and vertical shifts.3. The parameter a a a in part (a) represents a horizontal shift.4. The parameter b b b in part (b) represents a vertical shift.

STEP 2

1. Analyze and sketch the graph for y=(x+a)3 y = (x+a)^3 y=(x+a)3.2. Analyze and sketch the graph for y=x3−b y = x^3 - b y=x3−b.

STEP 3

The base function is y=x3 y = x^3 y=x3, which is a cubic function with an inflection point at the origin (0,0).

STEP 4

For y=(x+a)3 y = (x+a)^3 y=(x+a)3, the graph of y=x3 y = x^3 y=x3 is shifted horizontally to the left by a a a units because a>0 a > 0 a>0.

STEP 5

Sketch the graph of y=(x+a)3 y = (x+a)^3 y=(x+a)3: - Start with the graph of y=x3 y = x^3 y=x3. - Shift every point on this graph a a a units to the left. - The inflection point moves from (0,0) to (-a,0).

STEP 6

The base function is again y=x3 y = x^3 y=x3.

STEP 7

For y=x3−b y = x^3 - b y=x3−b, the graph of y=x3 y = x^3 y=x3 is shifted vertically downward by b b b units because b>0 b > 0 b>0.

STEP 8

Sketch the graph of y=x3−b y = x^3 - b y=x3−b: - Start with the graph of y=x3 y = x^3 y=x3. - Shift every point on this graph b b b units downward. - The inflection point moves from (0,0) to (0,-b). The graphs for each function have been sketched with the described transformations.

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Question2. Sketch the following graphs on separate diagrams. (a) $y=(x+a)^{3}, a>0$ (b) $y=x^{3}-b, b>0$

1. We are dealing with cubic functions, which have the general form $y = x^3$ .
2. The transformations involve horizontal and vertical shifts.
3. The parameter $a$ in part (a) represents a horizontal shift.
4. The parameter $b$ in part (b) represents a vertical shift.

1. Analyze and sketch the graph for $y = (x+a)^3$ .
2. Analyze and sketch the graph for $y = x^3 - b$ .

The base function is $y = x^3$ , which is a cubic function with an inflection point at the origin (0,0).

For $y = (x+a)^3$ , the graph of $y = x^3$ is shifted horizontally to the left by $a$ units because $a > 0$ .

Sketch the graph of $y = (x+a)^3$ : - Start with the graph of $y = x^3$ . - Shift every point on this graph $a$ units to the left. - The inflection point moves from (0,0) to (-a,0).

The base function is again $y = x^3$ .

For $y = x^3 - b$ , the graph of $y = x^3$ is shifted vertically downward by $b$ units because $b > 0$ .

Sketch the graph of $y = x^3 - b$ : - Start with the graph of $y = x^3$ . - Shift every point on this graph $b$ units downward. - The inflection point moves from (0,0) to (0,-b).
The graphs for each function have been sketched with the described transformations.