Parabola Definition: 3 Points (-5,-3), (-4,0), (0,-8)

Understanding the Parabola: Defining its Shape with Three Key Points

When we talk about a parabola, we're essentially discussing a specific type of curve that has a unique mathematical definition. At its core, a parabola is the set of all points in a plane that are equidistant from a fixed point (called the focus) and a fixed line (called the directrix). This elegant geometric property gives parabolas their characteristic U-shape, which can open upwards, downwards, sideways, or even at an angle. In the realm of algebra, parabolas are most commonly represented by quadratic equations. For instance, a standard parabola opening upwards or downwards can be described by an equation of the form $y = ax^2 + bx + c$, where 'a', 'b', and 'c' are constants. If the parabola opens sideways, the equation takes the form $x = ay^2 + by + c$. The coefficients 'a', 'b', and 'c' play crucial roles in determining the parabola's orientation, width, and position within the coordinate plane. The sign of 'a' dictates the direction of opening: if 'a' is positive, the parabola opens upwards (or to the right if it's a sideways parabola), and if 'a' is negative, it opens downwards (or to the left). The magnitude of 'a' influences how wide or narrow the parabola is; a larger absolute value of 'a' results in a narrower parabola, while a smaller absolute value leads to a wider one. The 'b' and 'c' terms, along with the 'a' term, influence the parabola's vertex and its overall placement on the graph. Finding the vertex is often a key step in analyzing a parabola, as it represents the minimum or maximum point of the function. For a parabola of the form $y = ax^2 + bx + c$, the x-coordinate of the vertex is given by $-b/(2a)$, and the y-coordinate is found by substituting this x-value back into the equation. For sideways parabolas ($x = ay^2 + by + c$), the roles of x and y are essentially swapped in the vertex formula. Understanding these algebraic representations allows us to not only graph parabolas accurately but also to solve problems involving them, such as finding intersections with lines or determining the trajectory of projectiles. The given points $(-5,-3)$, $(-4,0)$, and $(0,-8)$ provide us with concrete data that we can use to uncover the specific quadratic equation that defines the parabola passing through them.

Unlocking the Parabola's Equation: Using Three Points to Define the Curve

To define a parabola when you're given three distinct points, we leverage the general form of a quadratic equation. Let's assume our parabola opens either upwards or downwards, so its equation can be written as $y = ax^2 + bx + c$. The core idea here is that each of the given points must satisfy this equation. By substituting the coordinates of each point into this general form, we can set up a system of linear equations with three unknowns: 'a', 'b', and 'c'. This is where the magic of mathematics comes into play, allowing us to solve for these unknown coefficients and thus determine the unique parabola that passes through these specific locations. Let's take our given points: $(-5,-3)$, $(-4,0)$, and $(0,-8)$.

For the point $(-5,-3)$: Substituting $x = -5$ and $y = -3$ into $y = ax^2 + bx + c$, we get: $-3 = a(-5)^2 + b(-5) + c$ $-3 = 25a - 5b + c$ (Equation 1)

For the point $(-4,0)$: Substituting $x = -4$ and $y = 0$ into $y = ax^2 + bx + c$, we get: $0 = a(-4)^2 + b(-4) + c$ $0 = 16a - 4b + c$ (Equation 2)

For the point $(0,-8)$: Substituting $x = 0$ and $y = -8$ into $y = ax^2 + bx + c$, we get: $-8 = a(0)^2 + b(0) + c$ $-8 = c$ (Equation 3)

Now we have a system of three linear equations:

1. $25a - 5b + c = -3$
2. $16a - 4b + c = 0$
3. $c = -8$

From Equation 3, we already know the value of $c$. This is a significant simplification! We can now substitute $c = -8$ into Equations 1 and 2:

Substitute $c = -8$ into Equation 1: $25a - 5b - 8 = -3$ $25a - 5b = 5$ Dividing by 5, we get: $5a - b = 1$ (Equation 4)

Substitute $c = -8$ into Equation 2: $16a - 4b - 8 = 0$ $16a - 4b = 8$ Dividing by 4, we get: $4a - b = 2$ (Equation 5)

Now we have a system of two linear equations with two unknowns ('a' and 'b'): 4. $5a - b = 1$ 5. $4a - b = 2$

We can solve this system using several methods, such as substitution or elimination. Let's use elimination. We can subtract Equation 5 from Equation 4: $(5a - b) - (4a - b) = 1 - 2$ $5a - b - 4a + b = -1$ $a = -1$

Now that we have the value of 'a', we can substitute it back into either Equation 4 or Equation 5 to find 'b'. Let's use Equation 4: $5(-1) - b = 1$ $-5 - b = 1$ $-b = 6$ $b = -6$

So, we have found our coefficients: $a = -1$, $b = -6$, and $c = -8$. Substituting these values back into the general quadratic equation $y = ax^2 + bx + c$, we get the specific equation for the parabola that passes through the given three points: $y = -x^2 - 6x - 8$. This equation precisely defines the parabola that contains the points $(-5,-3)$, $(-4,0)$, and $(0,-8)$.

Analyzing the Parabola: Vertex, Axis of Symmetry, and Intercepts

Once we have the equation of a parabola, such as the one we found, $y = -x^2 - 6x - 8$, we can delve deeper into its characteristics. Understanding these features helps us visualize the parabola and its behavior on the coordinate plane. The vertex is a crucial point; it's the minimum or maximum point of the parabola. For an equation in the form $y = ax^2 + bx + c$, the x-coordinate of the vertex is given by the formula $x_v = -b/(2a)$. In our case, with $a = -1$ and $b = -6$, the x-coordinate of the vertex is: $x_v = -(-6) / (2 * -1)$ $x_v = 6 / -2$ $x_v = -3$

To find the y-coordinate of the vertex, we substitute this x-value back into our parabola's equation: $y_v = -(-3)^2 - 6(-3) - 8$ $y_v = -(9) + 18 - 8$ $y_v = -9 + 18 - 8$ $y_v = 9 - 8$ $y_v = 1$

Therefore, the vertex of our parabola is at $(-3, 1)$. Since the coefficient 'a' is negative (-1), the parabola opens downwards, meaning the vertex is indeed a maximum point.

The axis of symmetry is a vertical line that passes through the vertex and divides the parabola into two mirror-image halves. Its equation is simply $x = x_v$. For our parabola, the axis of symmetry is $x = -3$.

Intercepts are also important points to identify. The y-intercept is the point where the parabola crosses the y-axis, which occurs when $x = 0$. We already used this to find 'c'. Plugging $x = 0$ into our equation gives $y = -(0)^2 - 6(0) - 8 = -8$. So, the y-intercept is at $(0, -8)$, which was one of our given points.

The x-intercepts are the points where the parabola crosses the x-axis, which occurs when $y = 0$. To find these, we set our equation to zero and solve for x: $0 = -x^2 - 6x - 8$ We can multiply the entire equation by -1 to make the leading coefficient positive: $0 = x^2 + 6x + 8$ This is a quadratic equation that we can solve by factoring. We need two numbers that multiply to 8 and add to 6. These numbers are 2 and 4: $0 = (x + 2)(x + 4)$ Setting each factor to zero: $x + 2 = 0 ightarrow x = -2$ $x + 4 = 0 ightarrow x = -4$

So, the x-intercepts are at $(-2, 0)$ and $(-4, 0)$. Notice that $(-4,0)$ was another one of our given points.

By analyzing these components – the vertex, axis of symmetry, and intercepts – we gain a comprehensive understanding of the shape and position of the parabola defined by the points $(-5,-3)$, $(-4,0)$, and $(0,-8)$. This detailed analysis solidifies our definition and provides a clear picture of this specific parabolic curve.

For further exploration into the fascinating world of conic sections and their mathematical properties, you might find resources from Khan Academy incredibly helpful.