To find the equation of a quadratic function that passes through the given points $(0, 4)$, $(2, 14)$, and $(4, 8)$, we need to determine the quadratic function in the standard form:

$y = ax^2 + bx + c$

Step 1: Set up the system of equations

Substitute each point into the quadratic equation to get a system of equations:

1. For the point $(0, 4)$: $4 = a(0)^2 + b(0) + c \implies c = 4$

2. For the point $(2, 14)$: $14 = a(2)^2 + b(2) + c \implies 14 = 4a + 2b + c$

3. For the point $(4, 8)$: $8 = a(4)^2 + b(4) + c \implies 8 = 16a + 4b + c$

Step 2: Substitute $c = 4$ into the other equations

Now that we know $c = 4$, substitute this into the other two equations:

1. $14 = 4a + 2b + 4$ $14 - 4 = 4a + 2b \implies 10 = 4a + 2b$ $5 = 2a + b \quad \text{(Equation 1)}$

2. $8 = 16a + 4b + 4$ $8 - 4 = 16a + 4b \implies 4 = 16a + 4b$ $1 = 4a + b \quad \text{(Equation 2)}$

Step 3: Solve the system of linear equations

We have the following system: $2a + b = 5 \quad \text{(Equation 1)}$ $4a + b = 1 \quad \text{(Equation 2)}$

Subtract Equation 2 from Equation 1: $(2a + b) - (4a + b) = 5 - 1$ $-2a = 4 \implies a = -2$

Now substitute $a = -2$ back into Equation 1: $2(-2) + b = 5$ $-4 + b = 5 \implies b = 9$

Step 4: Write the final equation

Now that we have $a = -2$, $b = 9$, and $c = 4$, the quadratic function is:

$\boxed{y = -2x^2 + 9x + 4}$

Would you like more details on any step, or do you have any other questions?

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Related Questions:

1. How do you solve a system of linear equations with substitution?
2. What is the significance of the vertex in a quadratic function?
3. How can you determine the direction of the parabola (upward or downward)?
4. How do you find the roots of the quadratic equation?
5. What is the discriminant, and how does it relate to the number of real roots?

Tip: Always check your final quadratic equation by substituting the original points to ensure they satisfy the equation.