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The x-intercept of a graph is the point where the graph crosses the x-axis. To find the x-intercept, you need to set the y-value to zero in the equation. This is because at any x-intercept, the value of y is always zero. In the exercise given, the equation is \( y = x^4 - 25 \). By setting \( y = 0 \), the equation transforms to \( 0 = x^4 - 25 \). To solve for \( x \), rearrange the equation to \( x^4 = 25 \). You will then find \( x \) by taking the fourth root on both sides: \( x = \sqrt[4]{25} \). This operation gives two solutions, \( x = 5 \) and \( x = -5 \), due to the nature of even roots yielding both positive and negative values.
This means the graph of the equation will cross the x-axis at these points: \( (5, 0) \) and \( (-5, 0) \).
Remember, finding intercepts is a fundamental concept in graphing equations, as they provide key points where the graph interacts with the axes.

The y-intercept of a graph is where the line crosses the y-axis. For any y-intercept, the x-value is always zero. To find the y-intercept, simply substitute \( x = 0 \) into the equation. Using our initial equation, \( y = x^4 - 25 \), replace \( x \) with zero: \( y = (0)^4 - 25 \). The calculation simplifies to \( y = -25 \), which means that the y-intercept is at the point \( (0, -25) \).
Knowing the y-intercept is crucial in graphing, as it provides a starting point on the y-axis where the graph begins.

Solving equations is a step-by-step process to find unknown values that satisfy the equation. To find intercepts, you follow different steps depending on whether you're looking for the x- or y-intercept:

• To find an x-intercept, set \( y = 0 \) and solve for \( x \).
• To find a y-intercept, set \( x = 0 \) and solve for \( y \).

Finding intercepts often involves basic algebraic manipulation. In the given exercise, we focused on \( y = x^4 - 25 \). For the x-intercept, setting \( y = 0 \) leads to solving \( x^4 = 25 \). For the y-intercept, setting \( x = 0 \) directly yields \( y = -25 \).
Solving these equations requires understanding of square and fourth roots, algebraic rearrangements, and logical steps to approach finding solutions effectively.