Find The Y-Intercept Of A Parabola

Understanding the Y-Intercept

The y-intercept of a graph is a crucial point where the graph crosses the y-axis. In the context of a function, this occurs when the value of the independent variable (usually 'x') is zero. For the equation y = -rac{1}{2} x^2 + 2x + 3, we are looking for the specific point where the parabola intersects the vertical y-axis. This intersection happens at a single, unique point on the y-axis. Because the y-axis is defined by all points where $x=0$, any point on the y-axis will have an x-coordinate of 0. Therefore, to find the y-intercept, we simply need to substitute $x=0$ into the given equation and solve for 'y'. This process will yield the y-coordinate of the point where the graph crosses the y-axis. The resulting coordinate pair will always be in the form $(0, y)$, where 'y' is the calculated value. This fundamental concept applies to all functions and their graphs, not just parabolas. It's a foundational step in analyzing and understanding the behavior of mathematical functions. The visual representation of the y-intercept on a graph provides immediate insight into the function's starting point or baseline value when the input is zero. In many real-world applications, such as modeling physical phenomena or economic trends, the y-intercept often represents an initial condition or a fixed starting amount. For instance, in a linear equation representing cost, the y-intercept might signify the fixed costs incurred before any production begins. For a quadratic equation like the one presented, which describes a parabola, the y-intercept gives us a specific point on the curve that is easily identifiable and often significant in understanding the overall shape and position of the parabola relative to the y-axis. It's a simple yet powerful tool in the mathematician's toolkit.

Calculating the Y-Intercept for y = -rac{1}{2} x^2 + 2x + 3

To find the coordinates of the y-intercept for the given equation, y = -rac{1}{2} x^2 + 2x + 3, we follow the principle that the y-intercept occurs when $x=0$. Let's substitute $x=0$ into the equation:

y = -rac{1}{2} (0)^2 + 2(0) + 3

Now, we simplify the equation:

y = -rac{1}{2} (0) + 0 + 3

$y = 0 + 0 + 3$

$y = 3$

So, when $x=0$, $y=3$. This means the y-intercept is at the point $(0, 3)$. This calculation is straightforward and confirms that the graph of the parabola intersects the y-axis at the point where the y-coordinate is 3. The term $ax^2$ and $bx$ in the quadratic equation both evaluate to zero when $x=0$, leaving only the constant term, 'c'. In this equation, $c=3$, which directly gives us the y-intercept. This is a general rule for any quadratic equation in the standard form $y = ax^2 + bx + c$: the y-intercept is always $(0, c)$. This makes identifying the y-intercept a quick task once the equation is in standard form. It's one of the easiest points to determine on a parabola. Visualizing this on a graph, you would see the U-shaped curve of the parabola crossing the vertical y-axis precisely at the mark for 3. This point is vital for sketching the graph accurately and understanding its symmetry. The vertex, axis of symmetry, and x-intercepts are other key features, but the y-intercept is often the first point identified when analyzing a quadratic function. Its simplicity in calculation makes it an excellent starting point for more complex analyses of the parabola's behavior.

Analyzing the Options Provided

We are given several options for the coordinates of the y-intercept:

A. $\left(-\frac{1}{2}, 3\right)$ B. $(0,3)$ C. $(2,5)$ D. $(3,0)$

From our calculation, we found that the y-intercept occurs at $(0, 3)$. Let's examine why the other options are incorrect:

• Option A: $\left(-\frac{1}{2}, 3\right)$ - This point has an x-coordinate that is not 0. Therefore, it cannot be the y-intercept, as the y-intercept must lie on the y-axis where $x=0$.
• Option C: $(2,5)$ - This point has an x-coordinate of 2. To check if this point is on the graph, we can substitute $x=2$ into the equation: y = -rac{1}{2}(2)^2 + 2(2) + 3 = -rac{1}{2}(4) + 4 + 3 = -2 + 4 + 3 = 5. So, $(2,5)$ is a point on the parabola, but it is not the y-intercept because $x \neq 0$. This point is actually the vertex of the parabola.
• Option D: $(3,0)$ - This point has a y-coordinate of 0. This means it is an x-intercept (where the graph crosses the x-axis), not a y-intercept (where the graph crosses the y-axis).

Therefore, the correct coordinates of the y-intercept are $(0,3)$. This directly corresponds to option B.

The Significance of the Y-Intercept in Graphing

The y-intercept is more than just a calculated value; it's a fundamental characteristic that helps us visualize and understand the graph of an equation. For the parabola defined by y = -rac{1}{2} x^2 + 2x + 3, knowing that the y-intercept is at $(0, 3)$ gives us an anchor point. When sketching the graph, we know precisely where it will cross the vertical axis. This is especially helpful when combined with other key features of the parabola, such as the vertex and the axis of symmetry. The vertex represents the highest or lowest point of the parabola, and the axis of symmetry is the vertical line that divides the parabola into two mirror images. The y-intercept, along with the vertex, helps to orient the parabola correctly on the coordinate plane. If we were to plot the point $(0, 3)$, we would then look for other points to help define the curve. The fact that the parabola opens downwards (indicated by the negative coefficient of the $x^2$ term, -rac{1}{2}) and has its vertex at $(2, 5)$ (as determined in the previous section) means that the parabola will descend from the vertex on both sides. The y-intercept at $(0, 3)$ is to the left of the axis of symmetry ($x=2$), and there will be a corresponding point on the right side of the axis of symmetry that is at the same height (y=3). To find this point, we can use the symmetry. The x-coordinate of the y-intercept is 0, which is 2 units to the left of the axis of symmetry ($x=2$). So, there must be another point 2 units to the right of the axis of symmetry, at $x = 2 + 2 = 4$, with the same y-coordinate, $y=3$. Substituting $x=4$ into the equation confirms this: y = -rac{1}{2}(4)^2 + 2(4) + 3 = -rac{1}{2}(16) + 8 + 3 = -8 + 8 + 3 = 3. So, $(4,3)$ is another point on the parabola. This process of using the y-intercept and symmetry allows for a more accurate and informed sketch of the parabolic graph. In essence, the y-intercept provides the initial vertical positioning of the graph, making it an indispensable element in graphical analysis.

Conclusion: The Definitive Y-Intercept

In summary, to find the y-intercept of any function's graph, you must evaluate the function at $x=0$. For the quadratic equation y = -rac{1}{2} x^2 + 2x + 3, substituting $x=0$ directly yields $y=3$. Therefore, the coordinates of the y-intercept are $(0,3)$. This point is where the parabola crosses the y-axis, and it's a fundamental piece of information for understanding and sketching the graph. Remember that for any equation in the standard form $y = ax^2 + bx + c$, the y-intercept is always $(0, c)$. This makes identifying this specific point remarkably simple.

For further exploration into the properties of quadratic functions and graphing, you can refer to resources like Khan Academy's Algebra section or Math is Fun's explanation of parabolas.