To find the x and y intercepts of a quadratic function, you should first know what these intercepts mean.

The x-intercepts are the places where the graph cuts through the x-axis. This means that the y-value at these points is zero.

On the flip side, the y-intercept is where the graph touches the y-axis. This happens when the x-value is zero. Now, let’s go through the steps to find these intercepts.

Finding the Y-Intercept

To find the y-intercept of a quadratic function, you just need to replace ( x ) with 0 in the function.

For example, let's use the quadratic equation:

[ f(x) = ax^2 + bx + c ]

To find the y-intercept, you do:

[ f(0) = a(0)^2 + b(0) + c = c ]

So, the y-intercept is simply the constant term ( c ). This gives you the point ( (0, c) ) on the graph.

Example:
For the function ( f(x) = 2x^2 + 3x + 1 ), we find the y-intercept by doing:

[ f(0) = 2(0)^2 + 3(0) + 1 = 1 ]

This means the y-intercept is at the point ( (0, 1) ).

Finding the X-Intercepts

Now, let’s find the x-intercepts. This involves figuring out when ( f(x) = 0 ). So, we set the quadratic equation to zero:

[ ax^2 + bx + c = 0 ]

You can solve this using different methods, like factoring, completing the square, or using the quadratic formula.

The quadratic formula is:

[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ]

The part under the square root, ( b^2 - 4ac ), is called the discriminant. It helps us understand the number of x-intercepts we have.

• If the discriminant is positive (( b^2 - 4ac > 0 )), there are two different x-intercepts.
• If it’s zero (( b^2 - 4ac = 0 )), there is one x-intercept (the graph just touches the x-axis).
• If it’s negative (( b^2 - 4ac < 0 )), there are no real x-intercepts.

Example:
Let’s use the same function ( f(x) = 2x^2 + 3x + 1 ) to find the x-intercepts. We set it to zero:

[ 2x^2 + 3x + 1 = 0 ]

Now using the quadratic formula:

[ x = \frac{-3 \pm \sqrt{3^2 - 4(2)(1)}}{2(2)} ]

Calculating this gives:

[ x = \frac{-3 \pm \sqrt{9 - 8}}{4} ]

[ x = \frac{-3 \pm 1}{4} ]

Now, solving for the values, we get:

[ x = \frac{-2}{4} = -\frac{1}{2} \quad \text{and} \quad x = \frac{-4}{4} = -1 ]

So, the x-intercepts are at the points ( (-1, 0) ) and ( (-\frac{1}{2}, 0) ).

Summary

To sum it up, finding the intercepts of a quadratic function is pretty simple:

1. For the y-intercept, set ( x = 0 ) and solve for ( f(0) = c ).
2. For the x-intercepts, set ( f(x) = 0 ) and use any method you like—factoring, completing the square, or the quadratic formula.

With some practice, you’ll get really good at finding these important points on the graph of any quadratic function!