Problem 1 Find the zeros of the following ... [FREE SOLUTION]

Chapter 3: Problem 1

Find the zeros of the following functions graphically: (a) \(f(x)=x^{2}-8 x+15\) (b) \(g(x)=2 x^{2}-4 x-16\)

Short Answer

Expert verified

The zeros of \( f(x) = x^2 - 8x + 15 \) are \( x = 3 \) and \( x = 5 \), and the zeros of \( g(x) = 2x^2 - 4x - 16 \) are \( x = -2 \) and \( x = 4 \).

Step by step solution

Understand the Problem

The problem requires finding the zeros of two quadratic functions graphically. Zeros of a function occur where the graph intersects the x-axis.

Graph the Function f(x)

Graph the function \( f(x) = x^2 - 8x + 15 \). This is a parabola opening upwards. Find the points where the parabola crosses the x-axis to determine the zeros.

Identify Zeros of f(x)

Upon graphing \( f(x) = x^2 - 8x + 15 \), the x-axis intersections occur at \( x = 3 \) and \( x = 5 \). Therefore, the zeros of \( f(x) \) are \( x = 3 \) and \( x = 5 \).

Graph the Function g(x)

Graph the function \( g(x) = 2x^2 - 4x - 16 \). This is also a parabola opening upwards. Determine where this parabola intersects the x-axis.

Identify Zeros of g(x)

Upon graphing \( g(x) = 2x^2 - 4x - 16 \), the parabola intersects the x-axis at \( x = -2 \) and \( x = 4 \). Thus, the zeros of \( g(x) \) are \( x = -2 \) and \( x = 4 \).

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Key Concepts

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

Zeros of a Function

When we talk about the zeros of a function, we refer to the values of the variable that make the function equal to zero. In simpler terms, it's where the graph of the function meets or touches the x-axis. For a quadratic function such as \( f(x) = ax^2 + bx + c \), the zeros can be found by determining the points where \( f(x) = 0 \).

These points are crucial as they give insight into the roots or solutions of the quadratic equation.
In exercises involving quadratic functions, finding the zeros graphically provides a visual representation of the concept.

Understanding where a function crosses or touches the x-axis helps in comprehending the behavior and shape of its graph. It is integral to analyze these points whether solving algebraically or graphically.

Graphing Parabolas

Graphing a parabola involves plotting points that define the curving U-shape, characteristic of quadratic functions. The general form of a quadratic function is \( f(x) = ax^2 + bx + c \). Here's how to graph a parabola step-by-step:

Determine the direction: If \( a > 0 \), the parabola opens upwards. If \( a < 0 \), it opens downwards.
Find the vertex: The vertex is the highest or lowest point of the parabola, depending on its orientation. It can be calculated using the formula \((h, k)\), where \( h = -\frac{b}{2a} \) and \( k = f(h) \).
Identify the symmetry: Parabolas are symmetrical about the vertical line known as the axis of symmetry, which passes through the vertex.
Locate the zeros: Use the quadratic formula or factorization to find where the parabola intersects the x-axis.

Graphing provides a visual framework that helps in identifying these elements like the zeros and gives an intuitive understanding of the quadratic function's structure.

X-Axis Intersections

Intersections with the x-axis are essential points on the graph of any function, representing where the function's output is zero. For quadratic functions, these intersections are synonymous with the zeros of the function.

These intersections are vital because they represent the solutions to the equation \( ax^2 + bx + c = 0 \).
If a quadratic function has two distinct real zeros, its graph will intersect the x-axis at two points. If there is one real zero, the graph just touches the x-axis, which is called a repeated or double root.
If the quadratic has no real zeros, it means the parabola does not intersect the x-axis but rather lies entirely above or below it.

By identifying x-axis intersections, one can infer solutions and make predictions about the function's behavior.

Solving Quadratic Equations

Solving quadratic equations means finding the values of \( x \) that satisfy the equation \( ax^2 + bx + c = 0 \). There are several methods to solve these equations, each with its strengths:

Factoring: When possible, factoring the quadratic into simpler binomials makes solving straightforward as it leads directly to the zeros.
Quadratic Formula: Used universally for all quadratics and given by \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), it always provides the solutions (real or complex).
Completing the Square: This method involves reshaping the quadratic into a perfect square trinomial, then solving traditionally.

In addition to these, graphical methods involve plotting the quadratic and visually identifying its zeros where it intersects with the x-axis. Understanding these methods enriches problem-solving skills and highlights different avenues for reaching a solution.

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