To find the equation of a parabola with its axis parallel to the $x$-axis, we start by assuming a general form for the equation of the parabola:

$y = ax^2 + bx + c$

Since the axis is parallel to the $x$-axis, we rewrite it in terms of $y$ and $x$:

$x = ay^2 + by + c$

Now, let's substitute the points $(5, 4)$, $(11, 2)$, and $(21, -4)$ into this equation to form a system of equations:

1. For the point $(5, 4)$: $5 = a(4)^2 + b(4) + c$ Simplifying, we get: $5 = 16a + 4b + c$

2. For the point $(11, 2)$: $11 = a(2)^2 + b(2) + c$ Simplifying, we get: $11 = 4a + 2b + c$

3. For the point $(21, -4)$: $21 = a(-4)^2 + b(-4) + c$ Simplifying, we get: $21 = 16a - 4b + c$

Now, we have a system of three equations:

16a + 4b + c = 5 \\ 4a + 2b + c = 11 \\ 16a - 4b + c = 21 \end{cases}$$ Let's solve this system for $$a$$, $$b$$, and $$c$$. ### Step 1: Eliminate $$c$$ Subtract the second equation from the first to eliminate $$c$$: $$(16a + 4b + c) - (4a + 2b + c) = 5 - 11$$ $$12a + 2b = -6$$ Dividing by 2: $$6a + b = -3$$ Next, subtract the second equation from the third: $$(16a - 4b + c) - (4a + 2b + c) = 21 - 11$$ $$12a - 6b = 10$$ Dividing by 2: $$6a - 3b = 5$$ Now we have a simpler system with two equations: $$\begin{cases} 6a + b = -3 \\ 6a - 3b = 5 \end{cases}$$ ### Step 2: Solve for $$b$$ Subtract the first equation from the second: $$(6a - 3b) - (6a + b) = 5 + 3$$ $$-4b = 8$$ $$b = -2$$ ### Step 3: Substitute $$b$$ back to find $$a$$ Substitute $$b = -2$$ into $$6a + b = -3$$: $$6a - 2 = -3$$ $$6a = -1$$ $$a = -\frac{1}{6}$$ ### Step 4: Substitute $$a$$ and $$b$$ to find $$c$$ Substitute $$a = -\frac{1}{6}$$ and $$b = -2$$ into one of the original equations, say $$4a + 2b + c = 11$$: $$4\left(-\frac{1}{6}\right) + 2(-2) + c = 11$$ $$-\frac{2}{3} - 4 + c = 11$$ $$c = 11 + 4 + \frac{2}{3} = 15 + \frac{2}{3} = \frac{47}{3}$$ ### Final Equation Now, substituting $$a$$, $$b$$, and $$c$$ back into the equation $$x = ay^2 + by + c$$, we get: $$x = -\frac{1}{6}y^2 - 2y + \frac{47}{3}$$ This is the equation of the parabola that passes through the points $$(5, 4)$$, $$(11, 2)$$, and $$(21, -4)$$. Would you like further details on solving systems of equations or interpreting parabolas? --- 1. What are the main characteristics of a parabola with an axis parallel to the $$x$$-axis? 2. How can you verify if a given point lies on a parabola? 3. What role do the coefficients $$a$$, $$b$$, and $$c$$ play in shaping the parabola? 4. How would the solution change if the axis were parallel to the $$y$$-axis instead? 5. What methods exist for solving systems of equations involving quadratic terms? **Tip:** When solving a system of equations, try to eliminate variables systematically to simplify solving for each unknown.