Question

find the \\(x\\)- and \\(y\\)-intercepts of the graph. do not graph the equation.
\\y = x^3 - 25x\\
\\(x\\)-intercept \\((x, y) = \left(\quad\
ight)\\) (smallest \\(x\\)-value)
\\(x\\)-intercept \\((x, y) = \left(\quad\
ight)\\)
\\(x\\)-intercept \\((x, y) = \left(\quad\
ight)\\) (largest \\(x\\)-value)
\\(y\\)-intercept \\((x, y) = \left(\quad\
ight)\\)

Explanation:

⚡ Using what you learned: factoring and solving quadratic equations · 🆕 New: Finding x- and y-intercepts
To find where a graph crosses the axes, we set one variable to zero and solve for the other.

Step 1: Find the y-intercept

To find the \(y\)-intercept, set \(x = 0\) in the equation:

$$y = x^3 - 25x$$

$$y = (0)^3 - 25(0)$$

$$y = 0$$

So, the \(y\)-intercept is:

$$(x, y) = (0, 0)$$

Step 2: Find the x-intercepts

To find the \(x\)-intercepts, set \(y = 0\) and solve for \(x\):

$$0 = x^3 - 25x$$

Factor out the greatest common factor, \(x\):

$$0 = x(x^2 - 25)$$

Step 3: Factor completely and solve

Factor the difference of squares, \(x^2 - 25\):

$$0 = x(x - 5)(x + 5)$$

Set each factor to zero to find the solutions:

$$x = 0$$

$$x - 5 = 0 \implies x = 5$$

$$x + 5 = 0 \implies x = -5$$

The \(x\)-intercepts are:

$$(x, y) = (-5, 0), \quad (0, 0), \quad (5, 0)$$

Answer:

• \(x\)-intercept (smallest \(x\)-value): \((-5, 0)\)
• \(x\)-intercept: \((0, 0)\)
• \(x\)-intercept (largest \(x\)-value): \((5, 0)\)
• \(y\)-intercept: \((0, 0)\)