How to Find The X and Y Intercepts

 The x-intercepts are points where the graph of a function or an equation crosses or "touches" the x-axis of the Cartesian Plane. You may think of this as a point with y-value of zero. To find the x-intercepts of an equation, let y = 0 then solve for x. In point notation, it is written as ( x , 0 ) x-intercept of a line x-intercepts of a parabola The y-intercepts are points where the graph of a function or an equation crosses or "touches" the y-axis of the Cartesian Plane. You may think of this as a point with x-value of zero. To find the y-intercepts of an equation, let x = 0 then solve for y. In point notation, it is written as ( 0 , y ) y-intercept of a line y-intercept of a parabola

Example 1: From the graph, describe the x and y intercepts using point notation.

The graph crosses the x-axis at , therefore, we can write the x-intercepts as points .

Similarly, the graph crosses the y-axis at . Its y-intercept can be written as the point .

Example 2: Find the x and y intercepts of the line .

To find the x-intercepts algebraically, we let y = 0 in the equation and then solve for values of x. In the same manner, to find for y-intercepts algebraically, we let x = 0 in the equation and then solve for y.

 x-intercepts ( let y = 0, then solve for x ) y-intercepts ( let x = 0, then solve for y) written as point: written as point:

Here's the graph to verify that our answers are correct.

Example 3: Find the x and y intercepts of the quadratic equation .

The graph of this quadratic equation is a parabola. We expect it to have a "U" shape where it would either open up or down.

To solve for the x-intercept of this problem, you will factor a simple trinomial. Then you set each binomial factor equal to zero and solve for x.

 x-intercepts (let y = 0, then solve for x) y-intercepts (let x = 0, then solve for y) as points: as point:

Our solved values for both x and y intercepts match with the graphical solution.

Example 4: Find the x and y intercepts of the quadratic equation .

This is an example where the graph of the equation has a y-intercept but without an x-intercept.

• Let's find the y-intercept first because it's extremely easy! Plug in x = 0 then solve for y.

• Now for the x-intercept. Plug in y = 0, and solve for x.

The square root of a negative number is imaginary. This suggests that this equation does not have an x-intercept!

The graph can verify what's going on. Notice that the graph crossed the y-axis at (0,1); but never did with the x-axis.

Example 5: Find the x and y intercepts of the circle .

This is a good example to illustrate that it is possible for the graph of an equation to have x-intercepts but without y-intercepts.

 x-intercepts (let y = 0, then solve for x) y-intercepts (let x = 0, then solve for y) as points: none

When solving for y, we arrived at the situation of trying to get the square root of a negative number. The answer is imaginary, thus, no solution. That means, the equation doesn't have any y-intercepts.

The graph verifies that we are right for the values of our x-intercepts, and it has no y-intercepts.

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