It looks like you've uploaded an image from a worksheet titled "Maxima and Minima of Quadratic Functions 1". The first question from the image reads:

1. Given the quadratic function $y = ax^2 + bx + c$, the maximum value is 4 at $x = 1$, and the graph passes through the point $(3, 1)$. Find the values of the constants $a$, $b$, and $c$.

To solve this:

Step 1: Use the Maximum Condition

The maximum value occurs at $x = 1$, so:

• At $x = 1$, $y = 4$. Substituting this into the quadratic equation: $a(1)^2 + b(1) + c = 4$ This simplifies to: $a + b + c = 4 \quad \text{(Equation 1)}$

Step 2: Use the Point on the Graph

The quadratic also passes through the point $(3, 1)$, meaning:

• At $x = 3$, $y = 1$. Substituting into the quadratic equation: $a(3)^2 + b(3) + c = 1$ This simplifies to: $9a + 3b + c = 1 \quad \text{(Equation 2)}$

Step 3: Use the Maximum Slope Condition

At the maximum point, the derivative $y' = 0$ since the slope is zero. The derivative of $y = ax^2 + bx + c$ is: $y' = 2ax + b$ At $x = 1$, the slope is zero, so: $2a(1) + b = 0$ This simplifies to: $2a + b = 0 \quad \text{(Equation 3)}$

Step 4: Solve the System of Equations

Now we have three equations:

1. $a + b + c = 4$
2. $9a + 3b + c = 1$
3. $2a + b = 0$

We can solve this system of equations to find $a$, $b$, and $c$.

Would you like me to continue with solving the system, or do you have any specific part you'd like to focus on?

5 Related Questions:

1. How do we solve a system of equations with substitution or elimination?
2. What is the significance of the derivative being zero at a maximum or minimum point?
3. How does the shape of a parabola relate to its coefficient $a$?
4. Can a quadratic function have both a maximum and minimum value?
5. How do changes in the constants $b$ and $c$ affect the position of a parabola?

Tip:

Always check the condition of the slope being zero at maxima or minima when dealing with quadratic functions!