The intercepts of a function are the values of x when f(x) = 0 and the value of f(x) when x = 0, corresponding to the coordinate values of x and y where the graph of the function crosses the x- and y-axes. Find the y-intercept of a rational function as you would for any other type of function: plug in x = 0 and solve. Find the x-intercepts by factoring the numerator. Remember to exclude holes and vertical asymptotes when finding the intercepts.
Plug the value x = 0 into the rational function and determine the value of f(x) to find the y-intercept of the function. For example, plug x = 0 into the rational function f(x) = (x^2 - 3x + 2) / (x - 1) to get the value (0 - 0 + 2) / (0 - 1), which is equal to 2 / -1 or -2 (if the denominator is 0, there is a vertical asymptote or hole at x = 0 and therefore no y-intercept). The y-intercept of the function is y = -2.
Factor the numerator of the rational function completely. In the above example, factor the expression (x^2 - 3x + 2) into (x - 2)(x - 1).
Set the factors of the numerator equal to 0 and solve for the value of the variable to find the potential x-intercepts of the rational function. In the example, set the factors (x - 2) and (x - 1) equal to 0 to get the values x = 2 and x = 1.
Plug the values of x you found in Step 3 into the rational function to verify that they are x-intercepts. X-intercepts are values of x that make the function equal to 0. Plug x = 2 into the example function to get (2^2 - 6 + 2) / (2 - 1), which equals 0 / -1 or 0, so x = 2 is an x-intercept. Plug x = 1 into the function to get (1^2 - 3 + 2) / (1 - 1) to get 0 / 0, which means there is a hole at x = 1, so there is only one x-intercept, x = 2.
Finding x-intercepts
Points where the graph cross the x-axis are the x-intercepts, wherey = 0. The x-intercepts of a function are also known as the zeros or real roots of the corresponding equation. In the case of rational functions, x-intercepts exist when the numerator equals zero.
To determine the x-intercepts of the function, set the numerator equal to zero and solve for x. Eliminate the excluded values because the function will be undefined and will not cross the x-axis.
Example 1 Find the x-intercepts of the function.
Step 1. Simplify the function completely.
Since x = -4 is an excluded value, it can not be an x-intercept.
Step 2. Using the Zero Product Property, set the factor(s) in the numerator equal to zero and solve.
x-intercepts: (2, 0) or (1, 0)
To find an x intercept, substitute y = 0 into the equation and solve for x. To find the y axis intercept, substitute x = 0 into the equation and solve for y. For
example, find the ๐ฅ and y intercepts of 2y + 3๐ฅ = 12. To find the ๐ฅ-axis intercept, first substitute y = 0 into the equation. When y = 0, the equation 2y + 3๐ฅ = 12 becomes 3๐ฅ = 12. Solve the resulting equation for ๐ฅ. 3๐ฅ = 12 can be solved for ๐ฅ by dividing both sides of the equation by 3. ๐ฅ = 4 and so, the ๐ฅ-axis intercept has
coordinates (4, 0). To find the y-axis intercept, first substitute ๐ฅ = 0 into the equation. When ๐ฅ = 0, the equation 2y + 3๐ฅ = 12 becomes 2y = 12. Solve the resulting equation for y. 2y = 12 can be solved for y by dividing both sides of the equation by 2. y = 6 and so, the y-axis intercept has coordinates (0, 6) The
๐ฅ intercept is the coordinate where a graph touches or crosses through the ๐ฅ-axis. It has a y coordinate of 0. The y intercept is the coordinate where a graph touches or crosses the y-axis. It has an ๐ฅ coordinate of 0. The y-axis is the vertical axis that passes through the centre of the cartesian axes from bottom to top. It is marked with numbers known as y coordinates. The ๐ฅ-axis is the horizontal axis that passes through the centre of
the cartesian axes from left to right. It is marked with numbers known as ๐ฅ coordinates.How to Find and Graph x and y Axis Intercepts Video Lesson
What are ๐ฅ and y Intercepts
The y-axis intercept always has an ๐ฅ coordinate of 0. In the example shown above, the y intercept is (0, 5) because it passes through the y-axis at y = 5.
The ๐ฅ-axis intercept always has a y coordinate of 0. In the example shown above, the ๐ฅ intercept is (8, 0) because it passes through the ๐ฅ-axis at ๐ฅ = 8.
A function can only have one y-axis intercept. This is because a function can only have at most one output for any given input. When ๐ฅ = 0, a function can only have one output which is the y intercept value. The number of ๐ฅ-axis intercepts depends on the type of equation.
For example, in the quadratic equation shown above, there is only one y intercept at (0, 4), however, there are two ๐ฅ intercepts found at (1, 0) and (7, 0). There are two ๐ฅ-axis intercepts in a quadratic equation.
A relation can have an infinite number of ๐ฅ or y intercepts depending on the equation of the relation. For example, a circle equation can have 0, 1 or up to 2 ๐ฅ-axis and y-axis intercepts.
On the circle shown above, the y intercepts are marked at (0, -3) and (0, 5).
The ๐ฅ intercepts are marked at (-8, 0) and (2,0).
y intercepts always take the form (k, 0). They always have an ๐ฅ coordinate of 0.
๐ฅ intercepts always take the form (0, k). They always have a y coordinate of 0.
How to Graph A Line using x and y Intercepts
To graph a line using x and y intercepts:
- Substitute ๐ฅ=0 into the equation to find the y-intercept.
- Substitute y=0 into the equation to find the ๐ฅ-intercept.
- Connect these two intercepts with a straight line.
For example, graph the linear function of y โ 4๐ฅ = 8.
Step 1. Substitute ๐ฅ = 0 into the equation to find the y-intercept
When ๐ฅ = 0, the equation y โ 4๐ฅ = 8 becomes y = 8.
The y-intercept is therefore (0, 8)
Step 2. Substitute y = 0 into the equation to find the ๐ฅ-intercept
When y = 0, the equation y โ 4๐ฅ = 8 becomes -4๐ฅ = 8.
Dividing both sides by -4, we get ๐ฅ = -2.
The ๐ฅ-intercept is therefore (-2, 0)
Step 3. Connect these two intercepts with a straight line
The two intercepts are plotted at (-2, 0) and (0, 8).
A straight line is then drawn between these two points to complete the graph.
For example, the equation 3y + 3๐ฅ =6 is written in standard form. Find the ๐ฅ and y intercepts.
Here A = 3, B = 3 and C = 6.
Setting ๐ฅ = 0, the equation 3y + 3๐ฅ = 6 becomes 3y = 6 and so the y-intercept is y = 2.
The coordinate of the y intercept is (0, 2).
We can see that C/B becomes 6/3 which equals 2.
Setting y = 0, the equation 3y + 3๐ฅ = 6 becomes 3๐ฅ = 6 and so, the ๐ฅ-intercept is ๐ฅ = 2.
The coordinate of the ๐ฅ intercept is (2, 0)
We can see that C/A becomes 6/3 which equals 2.
This standard form equation can now be graphed by plotting these two intercept coordinates and drawing a line between them.
If the ๐ฅ and y axis intercepts are the same, the line has a gradient of -1. For every one unit right, the line travels one unit down.
Finding the ๐ฅ and y Intercepts with Fractions
To find the ๐ฅ intercept, substitute y=0 into the equation and solve for ๐ฅ. To find the y intercept, substitute ๐ฅ = 0 into the equation and solve for y. If there is a fraction following the substitution, multiply each term by the denominator and divide each term by the numerator to solve it.
For example, find the ๐ฅ and y intercepts of .
To find the ๐ฅ intercept, substitute y = 0 and solve for ๐ฅ.
This results in . Since there is a fraction, multiply by the denominator and then divide by the numerator.
Multiplying both sides of the equation by 3, the equation becomes 2๐ฅ = 12.
Then dividing both sides of the equation by 2, ๐ฅ = 6.
Therefore the ๐ฅ intercept is found at (6, 0).
To find the y intercept, substitute ๐ฅ = 0 and solve for y.
This results in . To find the intercept of this fractional equation, multiply both sides of the equation by the denominator of 2.
This results in 2y = 8.
Therefore the y intercept of this equation is (0, 8).
The line can be graphed by plotting the intercepts and drawing a line between them,
How to Find ๐ฅ and y Intercepts for a Linear Function
A linear equation is written in the form y = mx + b. b is the constant term and is the value of the y-intercept. The x-intercept is the value of x when y = 0. For a linear function, the x-intercept is equal to -b/m. For example, y = 2x โ 6 has a y-intercept of -6 and an x-intercept of 3.
In linear equations of the form, y = m๐ฅ + b, the value of m is the coefficient of ๐ฅ and b is the constant term. This means that m is the value ๐ฅ is multiplied by and b is the number on its own.
When written in slope-intercept form, the equation of a straight line is y = m๐ฅ + b.
To find the ๐ฅ intercept, set y = 0 and solve for x.
y = m๐ฅ + b becomes 0 = m๐ฅ + b.
We can rearrange this for ๐ฅ to get ๐ฅ = -b/m.
To find the y intercept, substitute ๐ฅ = 0 and solve for y.
y = m๐ฅ + b becomes y = b.
For example, in the equation y = 2๐ฅ โ 6, m = 2 and b = -6.
Therefore the y-axis intercept is b, which is -6. The y intercept is at (0, -6).
The ๐ฅ-axis intercept is -b/m, which is 6/2 which is 3. The ๐ฅ intercept is at (3, 0).
The same results for the ๐ฅ and y intercepts can be found by substituting y = 0 and ๐ฅ = 0 respectively into the equation y = 2๐ฅ โ 6.
Finding ๐ฅ and y Intercepts for Rational Functions
To find the x-axis intercept of a rational function, substitute y = 0 and solve for x. The x-axis intercept is therefore found when the numerator of the rational function equals zero. The y-axis intercept is found by substituting x = 0 into the function and evaluating the result.
For example, find the ๐ฅ and y intercepts for .
To find the ๐ฅ-axis intercept, set y = 0.
becomes .
We can multiply both sides of the equation by ๐ฅ + 1 to get . We can skip to this part of the solution when we are finding the ๐ฅ intercept of a rational function.
Simply set the numerator equal to zero.
Therefore 0 = (๐ฅ+2)(๐ฅ-2).
Setting each bracket equal to zero, the solutions become ๐ฅ = -2 and ๐ฅ = 2.
The ๐ฅ-intercepts are (-2, 0) and (2, 0).
To find the y-axis intercept, substitute ๐ฅ = 0 into the function.
becomes which becomes .
y = -4 and so, the y-axis intercept is (0, -4).
Finding ๐ฅ and y Intercepts for a Parabola
A parabola of the form y = ax2 + bx + c has only one y-axis intercept at (0, c). The parabola can have up to two x-axis intercepts which are its roots or zeros. To find the x-axis intercepts, set y = 0 and solve the quadratic equation using the quadratic formula or by factorisation.
For example, find the ๐ฅ and y intercepts of y = ๐ฅ2 โ 8x + 7.
The y-intercept can be found by substituting ๐ฅ = 0 into the equation. This results in y = 7.
More simply, the y-intercept is at (0, c). In the equation y = ๐ฅ2 โ 8x + 7, the value of c is 7. Therefore the y-axis intercept is at (0, 7).
To find the ๐ฅ-axis intercepts, we set y = 0 and solve for ๐ฅ.
y = ๐ฅ2 โ 8x + 7 becomes 0 = ๐ฅ2 โ 8x + 7. We can factorise the equation to get (๐ฅ โ 1)(๐ฅ โ 7) = 0.
Therefore, setting each bracket to equal 0, the solutions are ๐ฅ = 1 and ๐ฅ = 7. Therefore the ๐ฅ-axis intercepts are at (1, 0) and (7,0).
The quadratic formula can be used to find the ๐ฅ-axis intercepts of any parabola.
The quadratic formula tells us that . This means that the first ๐ฅ-axis intercept is found at and the second ๐ฅ-axis intercept is found at .
For the equation y = ๐ฅ2 โ 8x + 7: a = 1, b = -8 and c = 7.
The quadratic formula, becomes , which simplifies to , which results in ๐ฅ = 1 and ๐ฅ = 7.
For any quadratic function, the axis of symmetry is found exactly in between the ๐ฅ-axis intercepts. To find the axis of symmetry using the ๐ฅ-intercepts, simply add the ๐ฅ coordinates of each ๐ฅ-axis intercept and then divide this result by 2.
The two ๐ฅ intercepts are at ๐ฅ = 1 and ๐ฅ = 7. Adding 1 and 7 and then dividing by 2 gives us ๐ฅ = 4.
The equation of the axis of symmetry is ๐ฅ = 4.
The vertex is the turning point of a quadratic graph. The vertex of any quadratic, a๐ฅ2 + b๐ฅ + c lies on its axis of symmetry.
Therefore the ๐ฅ coordinate of the vertex is always exactly halfway between the two ๐ฅ-axis intercepts of the quadratic at . The y coordinate of the vertex can then be found by substituting this value of ๐ฅ into the original quadratic function.
For the equation, y = ๐ฅ2 โ 8x + 7, the equation for the ๐ฅ coordinate of the vertex becomes . This equals ๐ฅ = 4.
This means that the ๐ฅ coordinate of the vertex is 4.
To find the y coordinate of the vertex, simply substitute the ๐ฅ coordinate of the vertex into the original quadratic equation.
๐ฅ2 โ 8x + 7 is equal to -9 when ๐ฅ = 4.
Therefore the coordinates of the vertex are (4, -9).
Finding ๐ฅ and y Intercepts for a Circle
To find the x-intercepts of a circle, substitute y = 0 and solve the resulting quadratic for x. To find the y-intercepts of a circle, substitute x = 0 and solve the resulting quadratic for y. A circle may have 0, 1 or 2 x-axis or y-axis intercepts depending on the number of solutions to the quadratic.
For example, find the ๐ฅ and y intercepts of (๐ฅ+3)2 + (y-1)2 = 25.
To find the ๐ฅ intercept, substitute y = 0 to get (๐ฅ+3)2 + (0-1)2 = 25.
This becomes (๐ฅ+3)(๐ฅ+3) + (-1)2 = 25.
Expanding this, we get ๐ฅ2 + 6๐ฅ + 9 + 1 = 25. We set a quadratic equation equal to zero to solve it.
We get ๐ฅ2 + 6๐ฅ โ 15 = 0. This cannot be factorised but solving this with the quadratic formula we get ๐ฅ = -7.90 or ๐ฅ = 1.90.
To find the y intercepts of a circle, set ๐ฅ = 0 and solve the resulting quadratic equation for y.
(๐ฅ+3)2 + (y-1)2 = 25 becomes (0+3)2 + (y-1)2 = 25.
This becomes (3)2 + (y-1)(y-1) = 25 which can be expanded to get 9 + y2 โ 2y + 1 = 25.
Setting this quadratic equation equal to zero, we get y2 โ 2y โ 15 = 0.
This can be factorised to get (y-5)(y+3) = 0, which gives us the solutions of y =5 or y = -3.
These ๐ฅ and y intercepts are shown on the graph of the circle below.
๐ฅ and y Intercepts From a Table
A table of x and y values make up pairs of coordinates. The x-intercept is found from the row in the table with a y coordinate of 0. The y-intercept is found from the row in the table with an x coordinate of 0.
The table below shows the table of coordinates formed from the function y = 2๐ฅ โ 4.
The y-axis intercept is seen to be (0, -4). This is the only pair of coordinates that have an ๐ฅ value of 0.
The ๐ฅ-axis intercept is seen to be (2, 0). This is the only pair of coordinates that have a y value of 0.
How to Find the x and y Intercepts from 2 Points
To find the x and y intercepts from 2 points, first find the equation of the line. The x intercept can be found by substituting y = 0 into the equation of the line. The y intercept can be found by substituting x = 0 into the equation of the line.
Finding the y Intercept From 2 Points
To find the y intercept from 2 points:
- Find the gradient of the line by dividing the difference in the y coordinates by the difference in x coordinates.
- Substitute this gradient, m into the equation y=mx+c along with the x and y values of one of the coordinates.
- Use these values to work out c, which is the value of the y-intercept.
For example, find the y intercept of the line passing through (2, 3) and (4, 9).
Step 1. Find the gradient by dividing the change in y coordinates by the change in x coordinates.
Between the y coordinates of 3 and 9 there is a change of +6.
Between the x coordinates of 2 and 4 there is a change of +2.
6 รท 2 = 3 and so the gradient = 3.
Step 2. Substitute the gradient, m into the equation y = mx + c along with the x and y values of one of the coordinates.
We call the gradient m. Therefore as calculated in step 1, m = 3.
We now select the x and y values from either coordinate. We will choose (2, 3) so x = 2 and y = 3.
Substituting m = 3, x = 2 and y = 3 into y = mx + c,
we get 3 = 6 + c.
Step 3. Use these values to work out c, the y-intercept.
Since 3 = 6 + c, the value of c = -3.
Therefore the y intercept is y = -3.
The y-intercept is (0, -3).
Finding the x Intercept from 2 Points
To find the x intercept from 2 points:
- Find the equation of the line using the two points.
- Substitute y=0 into the equation of the line.
- Solve the resulting equation for x.
Step 1. Find the equation of the line using the 2 points.
As seen in the steps above, the equation of the line is y = 3x โ 3.
Step 2. Substitute y=0 into the equation of the line.
y = 3x โ 3 becomes 0 = 3x โ 3.
Step 3. Solve the resulting equation for x.
0 = 3x โ 3 can be solved by adding 3 to both sides.
3 = 3x
We divide both sides by 3 to get x = 1.
The x intercept is found at (1, 0).