One day jasmine earned $25.20 for 6 hours of work. Another day she earned $16.80 for 4 hours of work. Which function shows how much Jasmine earns d for working hours h hours?

Answer 1

To find a function that represents Jasmine's earnings based on her working hours, we can first determine her hourly wage from the two days of work provided.

1. First Day:

   • Earnings: $25.20
   • Hours: 6
   • Hourly Wage = Earnings / Hours = \( \frac{25.20}{6} = 4.20 \) dollars per hour

2. Second Day:

   • Earnings: $16.80
   • Hours: 4
   • Hourly Wage = Earnings / Hours = \( \frac{16.80}{4} = 4.20 \) dollars per hour

From both days, we see that Jasmine's hourly wage is consistently $4.20 per hour.

Now, we can express her total earnings \( d \) as a function of the hours worked \( h \). The relationship can be described by the function:

\[ d(h) = 4.20h \]

Where:

• \( d(h) \) is the amount Jasmine earns for \( h \) hours of work,
• \( 4.20 \) is her hourly wage,
• \( h \) is the number of hours worked.

So the function that shows how much Jasmine earns based on her working hours is:

\[ d(h) = 4.20h \]

Answer 2

which descibes the graph of y = - x^2 + 4?

Answer 3

The equation \( y = -x^2 + 4 \) describes a quadratic function. Here are the key features of the graph:

1. Shape: The graph is a downward-opening parabola because the coefficient of \( x^2 \) (which is -1) is negative.

2. Vertex: The vertex of the parabola can be found using the standard form of the quadratic equation. The vertex form for a parabola described by \( y = ax^2 + bx + c \) is given by \( (h, k) \), where \( h = -\frac{b}{2a} \) and \( k = f(h) \).

   In this case, the equation can be rewritten as:

   • Here, \( a = -1 \) and \( b = 0 \), so \( h = -\frac{0}{2(-1)} = 0 \).

   Substitute \( h = 0 \) back into the equation to find \( k \):

   • \( k = -0^2 + 4 = 4 \).

   Therefore, the vertex of the parabola is at the point \( (0, 4) \).

3. Y-intercept: The y-intercept occurs when \( x = 0 \):

   • \( y = -0^2 + 4 = 4 \). Thus, the graph intersects the y-axis at \( (0, 4) \).

4. X-intercepts: The x-intercepts occur when \( y = 0 \):

   • Set the equation to zero: \( 0 = -x^2 + 4 \).
   • Rearranging gives \( x^2 = 4 \), leading to \( x = \pm 2 \). Thus, the x-intercepts are at the points \( (-2, 0) \) and \( (2, 0) \).

5. Axis of symmetry: The axis of symmetry for the parabola is the vertical line that goes through the vertex. Since the vertex is at \( (0, 4) \), the axis of symmetry is the line \( x = 0 \) (the y-axis).

Summary:

• The graph is a downward-opening parabola.
• The vertex is at \( (0, 4) \).
• The y-intercept is \( (0, 4) \).
• The x-intercepts are \( (-2, 0) \) and \( (2, 0) \).
• The axis of symmetry is \( x = 0 \).

This parabola will look like a "∩" shape, with the highest point at the vertex (0, 4).

Answer 4

3a + 2b - 8a + b

Answer 5

To simplify the expression \(3a + 2b - 8a + b\), you can combine like terms:

1. Combine the \(a\) terms: \[ 3a - 8a = -5a \]

2. Combine the \(b\) terms: \[ 2b + b = 3b \]

Putting it all together, the simplified expression is: \[ -5a + 3b \]

So, \(3a + 2b - 8a + b\) simplifies to \(-5a + 3b\).

Answer 6

Which function is described by the values in the table?
x y
1 8
3 6
5 4
7 2

Answer 7

To find a function that fits the values provided in the table, let's analyze the pairs of \(x\) and \(y\):

\[ \begin{array}{c|c} x & y \ \hline 1 & 8 \ 3 & 6 \ 5 & 4 \ 7 & 2 \ \end{array} \]

Looking at the values of \(y\) in relation to \(x\), we can observe a pattern:

• As \(x\) increases, \(y\) decreases.
• The changes in \(y\) can be calculated:
  • From \(y = 8\) at \(x = 1\) to \(y = 6\) at \(x = 3\), the change is \(-2\).
  • From \(y = 6\) at \(x = 3\) to \(y = 4\) at \(x = 5\), the change is \(-2\).
  • From \(y = 4\) at \(x = 5\) to \(y = 2\) at \(x = 7\), the change is \(-2\).

This suggests a linear relationship between \(x\) and \(y\) with a constant decrease of \(2\) in \(y\) for every \(2\) increase in \(x\).

To find the equation of the line, we can use the two-point form of the line equation:

1. Identify two points: let's take \( (1, 8) \) and \( (3, 6) \).

2. Calculate the slope \(m\): \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{6 - 8}{3 - 1} = \frac{-2}{2} = -1 \]

3. Now, use the point-slope form \(y - y_1 = m(x - x_1)\) with point \((1, 8)\): \[ y - 8 = -1(x - 1) \]

4. Simplify the equation: \[ y - 8 = -x + 1 \ y = -x + 9 \]

Now let's verify if this equation holds for the other points:

• For \(x = 3\): \[ y = -3 + 9 = 6 \quad (\text{matches}) \]

• For \(x = 5\): \[ y = -5 + 9 = 4 \quad (\text{matches}) \]

• For \(x = 7\): \[ y = -7 + 9 = 2 \quad (\text{matches}) \]

Thus, the function that is described by the given values in the table is:

\[ y = -x + 9 \]

Answer 8

The sequence below shows the number of bacteria Arjun observed each hour for a science experiment:

5, 20, 80, 320, 1,280 , , ,

Which recursive funtion describes the number of bacteria observed at the nth hour?

Answer 9

To find a recursive function that describes the sequence of bacteria Arjun observed each hour, we need to analyze how the sequence progresses:

The given sequence is:

• Hour 0: \( 5 \)
• Hour 1: \( 20 \)
• Hour 2: \( 80 \)
• Hour 3: \( 320 \)
• Hour 4: \( 1,280 \)

Let’s identify the pattern:

1. Ratio of Consecutive Terms:
   • From \( 5 \) to \( 20 \): \( 20 \div 5 = 4 \)
   • From \( 20 \) to \( 80 \): \( 80 \div 20 = 4 \)
   • From \( 80 \) to \( 320 \): \( 320 \div 80 = 4 \)
   • From \( 320 \) to \( 1,280 \): \( 1,280 \div 320 = 4 \)

This shows that each term is multiplied by \( 4 \) to get the next term.

2. Establishing the Recursive Formula:
   • Let \( a_n \) be the number of bacteria observed at the \( n \)-th hour.
   • The relationship can be expressed as: \[ a_n = 4 \cdot a_{n-1} \]
   • The initial condition is: \[ a_0 = 5 \]

Final Recursive Function

So the recursive function that describes the number of bacteria observed at the \( n \)-th hour is:

\[ \begin{cases} a_n = 4 \cdot a_{n-1}, & \text{for } n \geq 1 \ a_0 = 5 & \end{cases} \]

This means that to get the number of bacteria for hour \( n \), you multiply the number of bacteria at the previous hour \( n-1 \) by \( 4 \), starting with \( 5 \) bacteria at hour \( 0 \).