Find the x and y Intercepts f(x)=(x^2-5)(x+6)
Brief explanation of the question:
The question is asking to determine the points at which the graph of the polynomial function \( f(x) = (x^2 - 5)(x + 6) \) intersects with the x-axis and the y-axis. The x-intercept(s) occur where the function equals zero, \( f(x) = 0 \), which typically means solving for the values of x that make the entire expression equal to zero (where the graph crosses the x-axis). The y-intercept occurs where the graph crosses the y-axis, which is found by evaluating the function at \( x = 0 \), namely \( f(0) \).
$f \left(\right. x \left.\right) = \left(\right. x^{2} - 5 \left.\right) \left(\right. x + 6 \left.\right)$
Answer
Solution:
Determine the x-intercepts
Step 1: Identify the x-intercepts
To locate the x-intercepts, let $y = 0$ and find the corresponding $x$ values.
Step 1.1: Set up the equation
$0 = (x^2 - 5)(x + 6)$
Step 1.2: Proceed to solve the equation
Step 1.2.1: Express the equation as
$(x^2 - 5)(x + 6) = 0$
Step 1.2.2: Apply the zero product property
$x^2 - 5 = 0$ or $x + 6 = 0$
Step 1.2.3: Solve for $x$ when $x^2 - 5 = 0$
Step 1.2.3.1: Set the first factor equal to zero
$x^2 - 5 = 0$
Step 1.2.3.2: Isolate $x^2$
Step 1.2.3.2.1: Add $5$ to both sides
$x^2 = 5$
Step 1.2.3.2.2: Extract the square root
$x = \pm\sqrt{5}$
Step 1.2.3.2.3: Consider both positive and negative roots
Step 1.2.3.2.3.1: Positive square root gives
$x = \sqrt{5}$
Step 1.2.3.2.3.2: Negative square root gives
$x = -\sqrt{5}$
Step 1.2.3.2.3.3: Combine both solutions
$x = \sqrt{5}, -\sqrt{5}$
Step 1.2.4: Solve for $x$ when $x + 6 = 0$
Step 1.2.4.1: Set the second factor equal to zero
$x + 6 = 0$
Step 1.2.4.2: Isolate $x$
$x = -6$
Step 1.2.5: Compile all $x$ values that satisfy the equation
$x = \sqrt{5}, -\sqrt{5}, -6$
Step 1.3: Express x-intercepts as coordinate points
x-intercepts: $(\sqrt{5}, 0), (-\sqrt{5}, 0), (-6, 0)$
Determine the y-intercepts
Step 2: Identify the y-intercepts
To find the y-intercepts, let $x = 0$ and solve for $y$.
Step 2.1: Set up the equation
$y = ((0)^2 - 5)(0 + 6)$
Step 2.2: Solve the equation
Step 2.2.1: Expand the equation
$y = (0^2 - 5)(0 + 6)$
Step 2.2.2: Simplify the equation
$y = (0 - 5)(0 + 6)$
Step 2.2.3: Further simplify
$y = (-5)(6)$
Step 2.2.4: Calculate the product
$y = -30$
Step 2.3: Express y-intercepts as coordinate points
y-intercept: $(0, -30)$
Compile the list of intercepts
x-intercepts: $(\sqrt{5}, 0), (-\sqrt{5}, 0), (-6, 0)$ y-intercept: $(0, -30)$
Knowledge Notes:
To solve for the x-intercepts of a function, we set $y = 0$ and solve the resulting equation for $x$. The x-intercepts are the points where the graph of the function crosses the x-axis. Similarly, to find the y-intercepts, we set $x = 0$ and solve for $y$. The y-intercepts are the points where the graph crosses the y-axis.
In this problem, we have a quadratic function multiplied by a linear function. The zero product property states that if a product of factors equals zero, at least one of the factors must be zero. We use this property to find the x-intercepts by setting each factor equal to zero and solving for $x$.
For the y-intercept, we substitute $x = 0$ into the function and simplify to find the value of $y$ when the graph crosses the y-axis.
The solutions to the equations provide the intercepts, which are expressed as coordinate points on the Cartesian plane.
