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Chapter 2: Problem 46

Find the \(x\) - and \(y\) -intercepts, if any, of the graph of the given and function \(f .\) Do not graph. $$ f(x)=(2 x-3)\left(x^{2}+8 x+16\right) $$

Short Answer

Expert verified
The y-intercept is (0, -48); x-intercepts are \(\left(\frac{3}{2}, 0\right)\) and (-4, 0).

Step by step solution

01

Find the y-intercept

To find the y-intercept, set \(x = 0\) and evaluate the function \(f(x)\). Substitute \(x = 0\) into the function: \(f(0) = (2(0)-3)((0)^2 + 8(0) + 16) = (-3)(16) = -48\). Thus, the y-intercept is at \((0, -48)\).
02

Find the x-intercept(s)

To find the x-intercept(s), set \(f(x) = 0\) and solve for \(x\). Since \((2x - 3)\) and \((x^2 + 8x + 16)\) are factors, set each factor to zero separately.
03

Solve the factor (2x - 3) = 0

Set \(2x - 3 = 0\) and solve for \(x\): \(2x = 3 \Rightarrow x = \frac{3}{2}\). Thus, one x-intercept is at \(\left(\frac{3}{2}, 0\right)\).
04

Solve the factor (x^2 + 8x + 16) = 0

Factor the quadratic as \((x + 4)^2 = 0\). Solve \((x + 4)^2 = 0\) to find \(x = -4\). This gives a repeated x-intercept at \((-4, 0)\).
05

List all intercepts

The y-intercept is at \((0, -48)\), and the x-intercepts are at \(\left(\frac{3}{2}, 0\right)\) and \((-4, 0)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding X-Intercepts
In mathematics, an x-intercept is the point where a graph crosses the x-axis. To find x-intercepts, we set the function equal to zero because, at these points, the output (or y-value) of the function is zero and crosses the x-axis. For instance, with the function \( f(x) = (2x - 3)(x^2 + 8x + 16) \), you need to solve \( f(x) = 0 \). This means we set the equation \((2x - 3)(x^2 + 8x + 16) = 0\) and solve for \(x\). You break it into factors and solve \(2x - 3 = 0\) and \(x^2 + 8x + 16 = 0\) separately. This gives the x-intercepts of the function, which are points on the graph where it touches the x-axis.
Discovering Y-Intercepts
A y-intercept, unlike the x-intercept, is where the graph crosses the y-axis. To find this, you simply set \(x = 0\) in the function and solve. The reason is that at the y-intercept, \(x\) is always zero. Thus, for our function \(f(x) = (2x - 3)(x^2 + 8x + 16)\), we substitute to get \(f(0)\). This calculation results in the y-intercept at the coordinate \((0, -48)\). It's a single point where the graph intersects the y-axis.
Factoring Quadratics Simplified
Factoring quadratics involves expressing the quadratic equation as a product of its factors. It's a critical step for solving equations and is especially useful when finding x-intercepts. The function given, \((x^2 + 8x + 16)\), is a perfect square trinomial. It can be simplified to \((x + 4)^2\). This means it has a repeated root, \(x = -4\), which gives us the x-intercept at this value. Understanding how to factor quadratics allows us to break down more complex equations into simpler parts and is essential for solving a variety of mathematical problems.
Solving Equations Effectively
Solving equations is about finding the value(s) of the variable that make the equation true. In our exercise, solving the equation \(f(x) = 0\) involves determining the x-values that make the product of factors zero. For \(f(x) = (2x - 3)(x^2 + 8x + 16)\), we factor each part and solve independently. The linear factor \(2x - 3 = 0\) solves to \(x = \frac{3}{2}\), and the quadratic \((x + 4)^2 = 0\) solves to \(x = -4\). Each solution corresponds to x-intercepts, showcasing how effective equation solving leads to understanding graph intersections.

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