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1. What is a Y Intercept?
The y intercept of a graph is the point where the graph intersects the y-axis. Any point on the y-axis has an x-coordinate of 0, meaning the x-coordinate of a y intercept is always 0. It’s a fundamental concept in algebra and calculus, helping to visualize functions.
Consider the line y = x + 3. This line crosses the y-axis at the point (0, 3). Therefore, (0, 3) is the y intercept of the line y = x + 3. This simple example illustrates how the y intercept serves as a starting point for graphing linear equations.
2. Y-Intercept Formula
The y intercept of a function represents the point where its graph meets the y-axis. As the x-coordinate of any point on the y-axis is always 0, we use this property to derive the formula for finding the y intercept. In essence, the y intercept is a point of the form (0, y). The formula to determine the y intercept is:
Y Intercept Formula
To find the y intercept of a function y = f(x), follow these steps:
- Substitute x = 0 into the equation.
- Solve for y.
- Express the y intercept as the point (0, y).
Here are a couple of examples to illustrate this:
- For y = 5x² + 2, substituting x = 0 yields y = 5(0)² + 2 = 2. Hence, the y intercept is (0, 2).
- For y = -5e^x, substituting x = 0 gives y = -5e⁰ = -5. Therefore, the y intercept is (0, -5).
3. Y-Intercept of a Straight Line
A straight line can be oriented horizontally, vertically, or at an angle. The y intercept of a horizontal line with the equation y = a is (0, a). A vertical line, however, does not have a y intercept because it never intersects the y-axis. Now, let’s explore how to find the y intercept of a straight line represented in various forms.
3.1. Y-Intercept in General Form
The general form of a straight line equation is ax + by + c = 0. To find the y intercept, substitute x = 0 and solve for y.
- a(0) + by + c = 0
- by + c = 0
- y = -c/b
Thus, the y intercept of a line in general form is (0, -c/b) or simply -c/b.
3.2. Y-Intercept in Slope-Intercept Form
The slope-intercept form of a line is given by y = mx + b. By definition, b represents the y intercept of the line. You can verify this by substituting x = 0 into y = mx + b to confirm that b is indeed the y intercept. Therefore, the y intercept of a line in slope-intercept form is (0, b) or just b.
3.3. Y-Intercept in Point-Slope Form
The point-slope form of a line equation is y – y₁ = m(x – x₁). To find the y intercept, substitute x = 0 and solve for y.
- y – y₁ = m(0 – x₁)
- y – y₁ = -mx₁
- y = y₁ – mx₁
Thus, the y intercept of a line in point-slope form is (0, y₁ – mx₁) or y₁ – mx₁.
4. How To Find Y-Intercept?
We’ve explored formulas for finding the y intercept of a line when its equation of the straight line is in different forms. However, it’s not always necessary to apply these formulas directly. For a polynomial function in the form y = a₁xⁿ + a₂xⁿ⁻¹ + … + aₙ, the y intercept is simply its constant term aₙ (or) (0, aₙ).
Simply substitute x = 0 in the equation of the line and solve for y. The resulting y value will be the y intercept, represented as y or (0, y).
| Equation of Line | Substitute x=0 and solve for y | y intercept |
|---|---|---|
| 3x+5y -6=0 | 3(0)+5y-6 =0 5y-6=0 y=6/5 | 6/5 (or) (0, 6/5) |
| y= 2x-3 | y=2(0)-3=-3 | -3 (or) (0, -3) |
The y intercept of a function can be easily determined by graphing it using a graphing calculator and identifying the point where the graph intersects the y-axis. A function has only one y intercept because otherwise, it would fail the vertical line test. The graph below illustrates the y intercept of the second equation from the table (y = 2x – 3).
5. Y-Intercept of a Quadratic Function (Parabola)
The procedure for finding the y intercept of a quadratic function or a parabola is similar to that of a line. If a quadratic equation is given, substitute x = 0 and solve for y to determine the y intercept.
Examples
| Equation of Parabola | Substitute x=0 and solve for y | y-Intercept |
|---|---|---|
| y= x² -2x -3 | y=0²-2(0)-3=-3 | -3 (or) (0, -3) |
| y= 2x²+5x-3 | y=2(0)²+5(0)-3=-3 | -3 (or) (0, -3) |
Important Notes on y-Intercept
- To find the y intercept, substitute x = 0 and solve for y.
- Similarly, to find the x-intercept, substitute y = 0 and solve for x.
- Lines parallel to the y-axis do not have y intercepts as they never intersect the y-axis.
- The y intercept is often used as a starting point when graphing a line by plotting two points.
- A function cannot have more than one y intercept.
Related Topics:
6. Examples on Y Intercept
Example 1: Find the y intercept of the following functions using the formula: a) y = x² – 3x + 2 b) y = (x² – 1) / x
Solution:
Using the y-intercept formula, substitute x = 0 in each function and solve for y.
a) For y = x² – 3x + 2:
- Substitute x = 0: y = 0² – 3(0) + 2 = 2
- The y intercept is (0, 2)
b) For y = (x² – 1) / x:
- Substitute x = 0: y = (0² – 1) / 0 = -1/0, which is undefined.
- The function does not have a y intercept.
Answer: a) (0, 2); b) Does Not Exist
Example 2: If the y intercept of the function y = a(x – 1)(x – 2)(x – 3) is (0, 12), find the value of “a”.
Solution:
The equation of the given function is y = a(x – 1)(x – 2)(x – 3). Use the y-intercept formula, which involves substituting x = 0.
- Substitute x = 0: y = a(0 – 1)(0 – 2)(0 – 3) = -6a
Thus, the y intercept is (0, -6a).
Given that the y intercept of the function is (0, 12), we have:
- -6a = 12
Divide both sides by -6:
- a = -2
Answer: a = -2
Example 3: If the y intercept of the function y = 3x² + ax + b is (0, -5), find the value of ‘b’.
Solution:
Using the y intercept formula, the y intercept of the given function is found by substituting x = 0.
- y = 3(0)² + a(0) + b = b
Thus, the y intercept is b.
Given that the y intercept is (0, -5) (or) -5, we have:
- b = -5
Answer: b = -5
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7. Practice Questions on y Intercept
8. FAQs on y Intercept
8.1. What is the Meaning of Y Intercept?
The y intercept of a graph is the point at which the graph intersects the y-axis. For a function, there can be at most one y intercept.
8.2. What is the Slope and Y Intercept of y = 6?
Comparing y = 6 with the slope-intercept form y = mx + b, we find:
- The slope, m = 0
- The y intercept, b = 6
8.3. How do you Find the Y-Intercept on the Graphing Calculator?
To find the y intercept using a graphing calculator:
- Press the “y=” button to enter the function.
- Press the “graph” button to display the graph of the function.
- Press the “trace” button and enter 0 to find the value of y when x is 0. This gives the y intercept.
8.4. What is Y Intercept Formula?
The y-intercept formula is used to find the point where the graph of a function intersects the y-axis. For a function y = f(x), the y intercept is found by substituting x = 0 and solving for y.
8.5. How to Derive Y Intercept Formula?
By definition, the y intercept is the point where a graph intersects the y-axis. Since the x-coordinate is 0 at every point on the y-axis, the y intercept of a function y = f(x) is found by substituting x = 0 and solving for y.
8.6. What are the Applications of Y Intercept Formula?
The y-intercept formula is used to find the y intercept of a function, which is essential in graphing the function.
8.7. Find the Y Intercept of the Graph Represented by the Equation x = y²+2y-3.
To find the y intercept, substitute x = 0 in the given equation and solve for y:
- y² + 2y – 3 = 0
- (y + 3)(y – 1) = 0
- y = -3 or y = 1
The y intercepts are (0, -3) and (0, 1).
8.8. How to Use the Y Intercept Formula in Finding the Y Intercept From a Graph?
The y-intercept formula tells us that the y intercept of a function y = f(x) is found by substituting x = 0. From a graph, the y intercept is the point where the graph intersects the y-axis, meaning it’s the point on the graph where the x-coordinate is 0.
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