The image asks to find a quadratic function $y = ax^2 + bx + c$ that passes through the points $(0,7), (2,13),$ and $(4,3)$.

To solve this, we can plug the points into the quadratic equation to form a system of equations:

1. For $(0, 7)$:
   $7 = a(0)^2 + b(0) + c$
   This simplifies to:
   $c = 7$

2. For $(2, 13)$:
   $13 = a(2)^2 + b(2) + c$
   Plugging in $c = 7$:
   $13 = 4a + 2b + 7$
   Simplifying:
   $4a + 2b = 6$
   (Equation 1)

3. For $(4, 3)$:
   $3 = a(4)^2 + b(4) + c$
   Plugging in $c = 7$:
   $3 = 16a + 4b + 7$
   Simplifying:
   $16a + 4b = -4$
   (Equation 2)

Now, solve the system of equations:

From Equation 1:
$4a + 2b = 6$
Dividing by 2:
$2a + b = 3$
(Equation 3)

From Equation 2:
$16a + 4b = -4$
Dividing by 4:
$4a + b = -1$
(Equation 4)

Now, subtract Equation 3 from Equation 4: $(4a + b) - (2a + b) = -1 - 3$ This simplifies to: $2a = -4 \quad \Rightarrow \quad a = -2$

Substitute $a = -2$ into Equation 3: $2(-2) + b = 3 \quad \Rightarrow \quad -4 + b = 3 \quad \Rightarrow \quad b = 7$

Now, we have $a = -2$, $b = 7$, and $c = 7$.

Thus, the quadratic equation is: $y = -2x^2 + 7x + 7$

Let me know if you need any further clarification or details! Here are a few related questions:

1. How do you derive the quadratic equation from three points?
2. How does solving a system of equations work?
3. Can a quadratic function have more than two solutions?
4. How would you graph the quadratic function $y = -2x^2 + 7x + 7$?
5. What are the properties of a parabola based on the quadratic equation?

Tip: The sign of the $a$-term in a quadratic equation determines whether the parabola opens upwards ($a > 0$) or downwards ($a < 0$).