91Ó°ÊÓ

When talking about x-intercepts, think about where a graph hits the x-axis on a coordinate plane. To find these points, set the value of the function, or "y," equal to zero. This is because on the x-axis, the height of the graph is zero. Let’s consider part (a) with the equation \( y = \sqrt{x^2 - 16} \). We set \( y = 0 \) and solve \( 0 = \sqrt{x^2 - 16} \). Eliminate the square root by squaring both sides, leading to \( x^2 - 16 = 0 \). Further, solve for \( x \) to find \( x = \pm 4 \), giving x-intercepts at \((4, 0)\) and \((-4, 0)\). For part (b), with \( y = \sqrt{64 - x^3} \), we again set \( y = 0 \). Solving it gives \( x^3 = 64 \) after squaring, leading to \( x = 4 \) after taking the cube root. Here, the x-intercept is \((4, 0)\). X-intercepts are pivotal as they show where the graph crosses or touches the x-axis.

Finding x-intercepts:

• Set \( y = 0 \).
• Solve the resulting equation for \( x \).
• Look for real number solutions to identify the intercepts.

Y-intercepts are another key concept that tells us where the graph crosses the y-axis. Here, x equals zero, as the graph’s horizontal location on the y-axis is zero. To find y-intercepts, we set \( x = 0 \) and solve for \( y \). For part (a), with equation \( y = \sqrt{x^2 - 16} \), substituting \( x = 0 \) results in \( y = \sqrt{-16} \), which isn't defined for real numbers, indicating no y-intercepts. But in part (b), substituting \( x = 0 \) in \( y = \sqrt{64 - x^3} \) simplifies to \( y = \sqrt{64} = 8 \). This means there is a y-intercept at \((0, 8)\).

Finding y-intercepts:

• Set \( x = 0 \).
• Solve the equation for \( y \).
• Check if \( y \) is a real number for intercepts.

Graphing equations is like creating a map of the equation onto a coordinate system. It helps visualize intersections and behavior of functions. A graph provides insight beyond numbers. For the equation \( y = \sqrt{x^2 - 16} \), you will notice the graph only touches the x-axis at points \((4, 0)\) and \((-4, 0)\), illustrating its restrictions and symmetry around the x-axis. No y-intercept means it doesn't cross the y-axis. The graph of \( y = \sqrt{64 - x^3} \) meets both axes, showing the x-intercept at \((4, 0)\) and y-intercept at \((0, 8)\). The shape and position of such functions are profoundly impacted by the roots inside the square root symbols.

When graphing:

• Identify intercepts as starting points.
• Plot key points and connect them smoothly.
• Consider symmetry and domain of the equation for correct layout.