How to use Intercepts

An intercept in algebra refers to the point at which a line or curve intersects with a coordinate axis. The \(x\)-intercept is the point at which the line or curve crosses the \(x\)-axis, and the \(y\)-intercept is the point at which it crosses the \(y\)-axis.

How to use Intercepts

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Step-by-step to use intercepts

To find out how to use intercepts, follow the step-by-step guide below:

  1. To find the \(x\)-intercept of a linear equation in slope-intercept form \((y=mx+b)\), set y equal to zero and solve for \(x\). This will give you the \(x\)-coordinate of the \(x\)-intercept.
  2. To find the \(y\)-intercept, set \(x\) equal to zero and solve for \(y\). This will give you the \(y\)-coordinate of the y-intercept.
  3. For example, the equation \(y=2x+1\) has an \(x\)-intercept at \((-0.5, 0)\) and a \(y\)-intercept at \((0, 1)\).
  4. To find the intercepts of a non-linear equation, you can set the equation equal to zero and solve for \(x\) or \(y\), depending on what type of intercept you are looking for. This may require factoring, completing the square, or using other algebraic techniques.

Using Intercepts – Example 1:

Find the \(x\)-intercept and \(y\)-intercept of the equation \(y=2x+1\).

Solution:

To find the \(x\)-intercept, we set \(y\) equal to zero and solve for \(x\):

\(y = 0 = 2x + 1\) →\(x = -0.5\)

So, the \(x\)-intercept is \((-0.5, 0)\).

To find the \(y\)-intercept, we set \(x\) equal to zero and solve for \(y\):

\(y=2(0)+1=1\)

So the \(y\)-intercept is \((0, 1)\).

Using Intercepts – Example 2:

Find the \(x\)-intercept and \(y\)-intercept of the equation \(y=-3x^2+4x -5\).

Solution:

To find the \(x\)-intercepts we set \(y\) equal to zero and solve for \(x\):

\(y = -3x^2 + 4x – 5 = 0\)→ \(-3x^2 + 4x – 5 = 0\) →\(x^2 – (\frac{4}{3})x + (\frac{5}{3}) = 0\)

To solve this equation we can factor it or use the quadratic formula. Factoring gives \((x – (\frac{5}{3}))(x – (\frac{4}{3}))=0\) so \(x=\frac{5}{3}\) or \(\frac{4}{3}\)

so the \(x\) intercepts are \((\frac{5}{3}, 0)\) and \((\frac{4}{3}, 0)\)

since it’s a parabola, it doesn’t have any \(y\)-intercept.

Exercises for Using Intercept

  1. Find the \(x\)-intercept and \(y\)-intercept of the equation \(3y=6x-9\).
  2. Find the \(x\)-intercept and \(y\)-intercept of the equation \(y=2x^2 + 7x – 8\).
This image has an empty alt attribute; its file name is answers.png
  1. \(\color{blue}{x-intercept=(\frac{3}{2}, 0), y-intercept= (0, -3)}\)
  2. \(\color{blue}{x-intercepts=(-4, 0) and (1, 0), y-intercept= (0, -8)}\)

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