91Ó°ÊÓ

X-intercepts are the points where a graph crosses the x-axis. These intercepts occur when the output variable, usually denoted as \( y \), equals zero. This means that at these points, the value of the dependent variable is zero.

• To find the x-intercepts: Set the equation equal to zero and solve for \( x \).

Let's use the example equation, \( y = 4x^2 - 1 \).
Starting by setting \( y = 0 \), we have:
\[ 0 = 4x^2 - 1 \]
By solving this equation, we add 1 to both sides, then divide by 4, leading to:
\[ 4x^2 = 1 \]
\[ x^2 = \frac{1}{4} \]
Taking the square root on both sides, we find two solutions for \( x \):
\[ x = \frac{1}{2} \] and \[ x = -\frac{1}{2} \]
Hence, the x-intercepts of the equation are \( (\frac{1}{2}, 0) \) and \( (-\frac{1}{2}, 0) \). These points indicate where the graph will touch or cross the x-axis.

Y-intercepts are the points where a graph crosses the y-axis. At these points, the independent variable \( x \) is zero. The y-intercept reveals the value of the function when \( x = 0 \).

• To find the y-intercepts: Substitute \( x = 0 \) into the equation and solve for \( y \).

For our sample quadratic equation \( y = 4x^2 - 1 \), substituting \( x = 0 \) gives:
\[ y = 4(0)^2 - 1 \]
This simplifies to:
\[ y = -1 \]
So, the y-intercept is at the point \( (0, -1) \).
This location indicates where the graph intersects the y-axis. The y-intercept gives a sense of the starting point of the graph on the vertical axis when viewed in a coordinate plane.

Quadratic equations are polynomial equations of degree 2. They are usually expressed in the standard form \( ax^2 + bx + c = 0 \). In our case, the given equation is \( y = 4x^2 - 1 \). This is a quadratic equation because it includes an \( x^2 \) term.

• Key features:
• The graph of a quadratic equation is a parabola, which could open upwards or downwards depending on the sign of the \( a \) coefficient.
• This specific equation, \( y = 4x^2 - 1 \), forms a parabola opening upwards since the \( a \) coefficient (4) is positive.
• The vertex of the parabola can be found using the vertex formula \( h = -\frac{b}{2a} \) for \( x \), but here \( b = 0 \), so the vertex is at \( x = 0 \).

Understanding quadratic equations is crucial because they frequently appear in various scientific computations and practical problems. Analyzing the x and y intercepts can help in understanding the behavior and trajectory of these parabolic graphs.