From the image you provided, I see that there are two tasks presented:

Task 4: Represent $P(x)$ on a graph
The function is given as:
$P(x) = -2x^2 + 80x - 600$ where $x$ represents the number of gadgets sold, and $P(x)$ represents the profit in dollars. The task involves determining:

1. The number of gadgets the company should sell to maximize profit.
2. The maximum profit value.
3. Graphical visualization of this function.

Task 3: Solve graphically
This involves solving a system of equations based on the problem:

• Amira buys 4 packs of colored pencils and 3 packs of markers for $25.
• Ben buys 2 packs of colored pencils and 5 packs of markers for $23.

The task is to determine the price of one pack of colored pencils and one pack of markers.

Let's solve each task step-by-step:

Task 4: Finding Maximum Profit and Graph

1. The given function $P(x) = -2x^2 + 80x - 600$ is a quadratic equation, and the coefficient of $x^2$ is negative, indicating it is a parabola opening downward.

2. To find the number of gadgets sold to maximize profit, use the vertex formula: $x = -\frac{b}{2a}$ Here, $a = -2$ and $b = 80$: $x = -\frac{80}{2(-2)} = 20$ Thus, selling 20 gadgets will maximize profit.

3. Substitute $x = 20$ into $P(x)$ to find the maximum profit: $P(20) = -2(20)^2 + 80(20) - 600$ $P(20) = -800 + 1600 - 600 = 200$ The maximum profit is $200.

4. Plot the graph for $P(x)$ over a reasonable range of $x$ values (e.g., $x = 0$ to $x = 40$).

Task 3: Solve Graphically

The system of equations is:

1. $4x + 3y = 25$
2. $2x + 5y = 23$

Using elimination or substitution:

1. Multiply the first equation by 2: $8x + 6y = 50$ Multiply the second equation by 4: $8x + 20y = 92$

2. Subtract the first modified equation from the second: $(8x + 20y) - (8x + 6y) = 92 - 50$ $14y = 42 \implies y = 3$

3. Substitute $y = 3$ into one of the original equations: $4x + 3(3) = 25$ $4x + 9 = 25 \implies 4x = 16 \implies x = 4$

Thus, the price of one pack of colored pencils is $4, and one pack of markers is $3.

4. To solve graphically, plot both equations and find their intersection point.

Would you like me to generate graphs for these tasks or provide further details?

Related Questions:

1. What is the significance of the vertex in a quadratic function's graph?
2. How can systems of equations be solved algebraically versus graphically?
3. Why does the coefficient of $x^2$ determine the shape of a parabola?
4. What other methods can be used to maximize profit in real-world scenarios?
5. How do graphing tools help in solving mathematical problems visually?

Tip:

Always double-check solutions by substituting back into the original equations to ensure accuracy.