Math / Algebra

Question5. The three equations provided all show the path of the same paper airplane thrown from the top of a building. State the three key features and explain what each key features tell us about the path of the paper airplane. Context is important! ( $h$ is height in metres and $t$ is time in seconds) $h=-2(t+3)(t-7) \quad h=-2 x^{2}+8 x+42 \quad h=-2(x-2)^{2}+50$

Studdy Solution

STEP 1

1. The equations represent the height $h$ of a paper airplane at time $t$ .
2. The equations are quadratic, indicating a parabolic path.
3. We need to identify and explain three key features of the path.

STEP 2

1. Identify the key features of the quadratic equations.
2. Interpret each feature in the context of the problem.
3. Explain what each feature tells us about the path of the paper airplane.

STEP 3

Identify the key features of the quadratic equations: - The roots (or zeros) of the equation. - The vertex of the parabola. - The direction of the parabola (opening up or down).

STEP 4

Interpret the roots of the equation: - From $h = -2(t+3)(t-7)$ , the roots are $t = -3$ and $t = 7$ . - These roots indicate the time at which the paper airplane is at ground level (height $h = 0$ ).

STEP 5

Interpret the vertex of the parabola: - From $h = -2(x-2)^{2}+50$ , the vertex is at $(t = 2, h = 50)$ . - The vertex represents the maximum height the paper airplane reaches, which is 50 meters at 2 seconds.

STEP 6

Interpret the direction of the parabola: - All equations have a leading coefficient of $-2$ , indicating the parabola opens downwards. - This tells us the paper airplane reaches a maximum height and then descends.

STEP 7

Explain what each feature tells us about the path of the paper airplane: - The roots $t = -3$ and $t = 7$ indicate the paper airplane is launched before $t = 0$ and lands at $t = 7$ seconds. - The vertex at $(2, 50)$ indicates the highest point in the flight, 50 meters above the ground, occurring 2 seconds after launch. - The downward opening parabola indicates the paper airplane follows a typical projectile motion, rising to a peak and then falling.
The three key features are:
1. Roots: $t = -3$ and $t = 7$ (launch and landing times).
2. Vertex: $(2, 50)$ (maximum height).
3. Direction: Downward opening (indicating projectile motion).

Was this helpful?

Studdy Solution

STEP 1

1. The equations represent the height $h$ of a paper airplane at time $t$ .
2. The equations are quadratic, indicating a parabolic path.
3. We need to identify and explain three key features of the path.

STEP 2

1. Identify the key features of the quadratic equations.
2. Interpret each feature in the context of the problem.
3. Explain what each feature tells us about the path of the paper airplane.

STEP 3

Identify the key features of the quadratic equations: - The roots (or zeros) of the equation. - The vertex of the parabola. - The direction of the parabola (opening up or down).

STEP 4

Interpret the roots of the equation: - From $h = -2(t+3)(t-7)$ , the roots are $t = -3$ and $t = 7$ . - These roots indicate the time at which the paper airplane is at ground level (height $h = 0$ ).

STEP 5

Interpret the vertex of the parabola: - From $h = -2(x-2)^{2}+50$ , the vertex is at $(t = 2, h = 50)$ . - The vertex represents the maximum height the paper airplane reaches, which is 50 meters at 2 seconds.

STEP 6

Interpret the direction of the parabola: - All equations have a leading coefficient of $-2$ , indicating the parabola opens downwards. - This tells us the paper airplane reaches a maximum height and then descends.

STEP 7

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Studdy Solution

STEP 1

1. The equations represent the height h h h of a paper airplane at time t t t.2. The equations are quadratic, indicating a parabolic path.3. We need to identify and explain three key features of the path.

STEP 2

1. Identify the key features of the quadratic equations.2. Interpret each feature in the context of the problem.3. Explain what each feature tells us about the path of the paper airplane.

STEP 3

Identify the key features of the quadratic equations: - The roots (or zeros) of the equation. - The vertex of the parabola. - The direction of the parabola (opening up or down).

STEP 4

Interpret the roots of the equation: - From h=−2(t+3)(t−7) h = -2(t+3)(t-7) h=−2(t+3)(t−7), the roots are t=−3 t = -3 t=−3 and t=7 t = 7 t=7. - These roots indicate the time at which the paper airplane is at ground level (height h=0 h = 0 h=0).

STEP 5

Interpret the vertex of the parabola: - From h=−2(x−2)2+50 h = -2(x-2)^{2}+50 h=−2(x−2)2+50, the vertex is at (t=2,h=50) (t = 2, h = 50) (t=2,h=50). - The vertex represents the maximum height the paper airplane reaches, which is 50 meters at 2 seconds.

STEP 6

Interpret the direction of the parabola: - All equations have a leading coefficient of −2-2−2, indicating the parabola opens downwards. - This tells us the paper airplane reaches a maximum height and then descends.

STEP 7

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. The equations represent the height $h$ of a paper airplane at time $t$ .
2. The equations are quadratic, indicating a parabolic path.
3. We need to identify and explain three key features of the path.

1. Identify the key features of the quadratic equations.
2. Interpret each feature in the context of the problem.
3. Explain what each feature tells us about the path of the paper airplane.

Interpret the roots of the equation: - From $h = -2(t+3)(t-7)$ , the roots are $t = -3$ and $t = 7$ . - These roots indicate the time at which the paper airplane is at ground level (height $h = 0$ ).

Interpret the vertex of the parabola: - From $h = -2(x-2)^{2}+50$ , the vertex is at $(t = 2, h = 50)$ . - The vertex represents the maximum height the paper airplane reaches, which is 50 meters at 2 seconds.

Interpret the direction of the parabola: - All equations have a leading coefficient of $-2$ , indicating the parabola opens downwards. - This tells us the paper airplane reaches a maximum height and then descends.