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Magazine Chapter 5: Problem 28
Solve each equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located. $$ x^{2}+4 x-4=0 $$
Short Answer
Expert verified
The roots are approximately between -4 and -1.
Step by step solution
01
Rewrite the Equation
We are given the equation \(x^2 + 4x - 4 = 0\). To solve it by graphing, rewrite the equation in the form of a function: \(f(x) = x^2 + 4x - 4\). This will allow us to graph \(y = f(x)\).
02
Identify the Vertex and Axis of Symmetry
The equation \(f(x) = x^2 + 4x - 4\) is a quadratic function, which graphs as a parabola. The vertex form of a quadratic is \(f(x) = a(x-h)^2 + k\), where \((h, k)\) is the vertex. We don't need the vertex form, but we can calculate the axis of symmetry using \(x = -\frac{b}{2a}\). Here, \(a = 1\) and \(b = 4\), so \(x = -\frac{4}{2 \cdot 1} = -2\).
03
Create and Plot a Table of Values
To draw the graph, we create a table of values around the vertex \(x = -2\), calculating \(f(x) = x^2 + 4x - 4\) at each value. Calculate for at least \(x = -3, -2, -1\), and a few values on either side if necessary.
04
Graph the Parabola
Plot the points calculated in the table from Step 3 on a coordinate system. Connect these points with a smooth curve to sketch the parabola.
05
Find the Roots on the Graph
The roots of the equation, or the x-values where the parabola intersects the x-axis (\(y=0\)), represent the solutions to \(x^2 + 4x - 4 = 0\). If the roots are integers, they will cross at these points on the graph. If not, observe the x-axis between which the parabola crosses.
06
Determine Exact or Approximate Roots
Based on the graph, identify the points where the parabola intersects the x-axis. If the exact integers are not evident, determine the consecutive integers between which the roots fall.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Graphing Parabolas
Graphing a parabola involves visually representing a quadratic function on a coordinate plane. A parabola is the U-shaped curve that represents a quadratic equation. For the equation \( f(x) = x^2 + 4x - 4 \), it can be understood as
- Opening Upward or Downward: The "a" value in the equation \( ax^2 + bx + c \) determines the direction of the parabola. If \( a > 0 \), the parabola opens upward. If \( a < 0 \), it opens downward. Here, \( a = 1 \), so it opens upward.
- Vertex: This is the highest or lowest point of a parabola. It can be found using the formula \( x = -\frac{b}{2a} \). For our equation, the vertex \( x \) coordinate is \( -2 \).
- Axis of Symmetry: This is the vertical line that divides the parabola into two mirror images. It is also given by \( x = -\frac{b}{2a} \), or \( x = -2 \) here.
Quadratic Functions
Quadratic functions are polynomial functions with the highest degree of 2. The standard form of a quadratic function is expressed as \( f(x) = ax^2 + bx + c \). Each term holds critical information:
- "a" Coefficient: Indicates the parabola's direction (upward or downward) and its "width." A larger absolute value of a makes for a "narrower" graph, while a smaller absolute value makes it "wider."
- "b" Coefficient: Plays a role in determining the axis of symmetry and affects the location of the vertex on the x-axis. For our equation, \( b = 4 \).
- "c" Constant: Determines the y-intercept, the point where the graph crosses the y-axis. Here, \( c = -4 \), meaning the graph crosses the y-axis at \( (0, -4) \).
Roots of Equations
The roots of a quadratic equation are solutions to \( ax^2 + bx + c = 0 \). These are the points where the parabola intersects the x-axis. For our function \( x^2 + 4x - 4 = 0 \), you can find the roots by:
- Graphing: Plot the quadratic function and identify x-values where the parabola crosses the x-axis. These are the roots. If exact x-values aren't easily seen, observe the integers the roots fall between.
- Factoring: Look for two numbers that multiply to "c" and add to "b," then set each factor to zero.
- Quadratic Formula: Use \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) if other methods do not suffice.
