Finding the y-coordinate of a point, particularly point I, involves understanding the context in which the point is situated. Usually, point I could be part of a geometric figure, a graph of a function, or a specific problem in coordinate geometry. Let’s explore a few common scenarios where you might need to find the y-coordinate of point I.
Scenario 1: Point I on a Line
If point I lies on a line with a known equation, you can find its y-coordinate using the line’s equation. For example, if the line’s equation is given as $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept, and you know the x-coordinate of point I, you can substitute the x-coordinate into the equation to find the y-coordinate.
Example
Suppose the line’s equation is $y = 2x + 3$ and the x-coordinate of point I is 4. Substitute $x = 4$ into the equation:
$y = 2(4) + 3$
$y = 8 + 3$
$y = 11$
So, the y-coordinate of point I is 11.
Scenario 2: Point I on a Parabola
If point I is on a parabola, the equation of the parabola is typically given in the form $y = ax^2 + bx + c$. Similar to the linear case, if you know the x-coordinate of point I, you can substitute it into the equation to find the y-coordinate.
Example
Consider the equation of the parabola $y = x^2 – 4x + 1$ and the x-coordinate of point I is 2. Substitute $x = 2$ into the equation:
$y = (2)^2 – 4(2) + 1$
$y = 4 – 8 + 1$
$y = -3$
So, the y-coordinate of point I is -3.
Scenario 3: Point I on a Circle
For a point on a circle, the equation is generally given in the form $(x – h)^2 + (y – k)^2 = r^2$, where $(h, k)$ is the center of the circle and $r$ is the radius. If you know the x-coordinate of point I, you can substitute it into the equation to solve for the y-coordinate. Note that you might get two values for y (positive and negative) since a circle is symmetric.
Example
Consider the circle with the equation $(x – 3)^2 + (y – 2)^2 = 25$ and the x-coordinate of point I is 6. Substitute $x = 6$ into the equation:
$(6 – 3)^2 + (y – 2)^2 = 25$
$3^2 + (y – 2)^2 = 25$
$9 + (y – 2)^2 = 25$
$(y – 2)^2 = 16$
$y – 2 = text{±}4$
$y = 6$ or $y = -2$
So, the y-coordinate of point I could be either 6 or -2.
Scenario 4: Point I as the Intersection of Two Lines
If point I is the intersection of two lines, you need to solve the system of equations representing the lines to find the coordinates of point I. Let’s say the equations of the lines are $y = m_1x + b_1$ and $y = m_2x + b_2$. Set the equations equal to each other and solve for x, then substitute back to find y.
Example
Consider the lines $y = 2x + 3$ and $y = -x + 1$. Set the equations equal:
$2x + 3 = -x + 1$
$3x = -2$
$x = -frac{2}{3}$
Now, substitute $x = -frac{2}{3}$ into either equation to find y. Using $y = 2x + 3$:
$y = 2bigg(-frac{2}{3}bigg) + 3$
$y = -frac{4}{3} + 3$
$y = -frac{4}{3} + frac{9}{3}$
$y = frac{5}{3}$
So, the y-coordinate of point I is $frac{5}{3}$
Scenario 5: Point I on a Function Graph
If point I is on the graph of a function, the y-coordinate can be found by substituting the x-coordinate into the function’s equation. For example, if the function is $f(x) = 3x^2 – 2x + 5$ and you know the x-coordinate of point I, substitute it into the function to find y.
Example
Consider the function $f(x) = 3x^2 – 2x + 5$ and the x-coordinate of point I is 1. Substitute $x = 1$ into the function:
$f(1) = 3(1)^2 – 2(1) + 5$
$f(1) = 3 – 2 + 5$
$f(1) = 6$
So, the y-coordinate of point I is 6.
Conclusion
Finding the y-coordinate of point I depends on the context of the problem. Whether point I is on a line, a parabola, a circle, or the intersection of two lines, the key is to use the given equations and substitute the known x-coordinate to solve for y. Understanding these methods will help you tackle a variety of problems in coordinate geometry and function graphs.