Problem 69
Question
Solve the quadratic equation and then use a graphing utility to graph the related quadratic function in the standard viewing window. Discuss how the graph of the quadratic function relates to the solutions of the quadratic equation. Function \(y=x^{2}+3 x-5\) Equation $$x^{2}+3 x-5=0$$
Step-by-Step Solution
Verified Answer
Solutions to the equation are \(x=1\) and \(x=-5\). The graph of related function \(y=x^{2}+3x-5\) intersects the x-axis at \(x=1\) and \(x=-5\), the solutions of the quadratic equation.
1Step 1: Understanding the Quadratic Formula
The quadratic formula is used to find the roots of a quadratic equation, which is an equation that can be written in the form \(ax^{2} + bx + c = 0\). The formula states that the roots of such an equation are given by \[x = \frac{-b ± \sqrt{b^{2}-4ac}}{2a}\]
2Step 2: Application of the Quadratic Formula
Let's apply the quadratic formula to our equation \(x^{2}+3 x-5=0\). In this case, a=1, b=3, c=-5. Substituting these into the quadratic formula gives us: \[x = \frac{-3 ± \sqrt{3^{2}-4*1*(-5)}}{2*1}\]. After simplifying, we find that the solutions to the equation are \[x = 1\] and \[x = -5\].
3Step 3: Graph the Quadratic Function
The related function is \(y=x^{2}+3 x-5\). To plot this on a graph, we can first plot the roots found above, \(x=1\) and \(x=-5\), on the x-axis. Next, we draw the general parabolic shape of a quadratic function, making sure it passes through these points. The graph opens upwards since the coefficient of \(x^{2}\) is positive.
4Step 4: Discuss the Relationship
We can see that the graph of the function \(y=x^{2}+3 x-5\) intersects the x-axis precisely at the points \(x=1\) and \(x=-5\). These are the solutions of the equation \(x^{2}+3 x-5=0\). Therefore, the roots of a quadratic equation correspond to where its graph intersects with the x-axis.
Key Concepts
Quadratic FormulaRoots of Quadratic EquationGraphing Quadratic Functions
Quadratic Formula
The quadratic formula is a robust tool for solving quadratic equations, which are equations of the form \(ax^2 + bx + c = 0\) where \(a\), \(b\), and \(c\) are coefficients and \(a \eq 0\). This formula provides a straightforward way to find the roots of a quadratic equation by using the coefficients of the terms.
The formula is: \[x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\] In essence, it is a blueprint that tells you how to find the points where the parabola representing the equation crosses the x-axis. The '±' indicates that there are two solutions (roots): one by adding the square root term and one by subtracting it.
Understanding how to effectively substitute the coefficients into the quadratic formula and perform the arithmetic is vital. When a student encounters \(x^2 + 3x - 5 = 0\), for example, they recognize \(a = 1\), \(b = 3\), and \(c = -5\). By substituting these values into the formula, they can find the solutions efficiently.
The formula is: \[x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\] In essence, it is a blueprint that tells you how to find the points where the parabola representing the equation crosses the x-axis. The '±' indicates that there are two solutions (roots): one by adding the square root term and one by subtracting it.
Understanding how to effectively substitute the coefficients into the quadratic formula and perform the arithmetic is vital. When a student encounters \(x^2 + 3x - 5 = 0\), for example, they recognize \(a = 1\), \(b = 3\), and \(c = -5\). By substituting these values into the formula, they can find the solutions efficiently.
Roots of Quadratic Equation
In quadratic equations, the term 'roots' refers to the values of \(x\) that make the equation equal to zero. These roots are crucial because they depict the values where the quadratic function intersects the x-axis on a graph. There are various methods to find these roots, one of which is the quadratic formula.
For the given equation \(x^2+3x-5=0\), when applying the quadratic formula, we get the roots \(x = 1\) and \(x = -5\), which are the points at which the related quadratic function will touch or cross the x-axis. Roots can be real or complex. If the discriminant \(b^2 - 4ac\) is positive, the roots are real and distinct; if it's zero, the roots are real and equal; and if negative, the roots are complex. This knowledge is not only essential for solving equations on paper but also for understanding the behavior of quadratic functions in real-life applications.
For the given equation \(x^2+3x-5=0\), when applying the quadratic formula, we get the roots \(x = 1\) and \(x = -5\), which are the points at which the related quadratic function will touch or cross the x-axis. Roots can be real or complex. If the discriminant \(b^2 - 4ac\) is positive, the roots are real and distinct; if it's zero, the roots are real and equal; and if negative, the roots are complex. This knowledge is not only essential for solving equations on paper but also for understanding the behavior of quadratic functions in real-life applications.
Graphing Quadratic Functions
Graphing is a powerful way to visualize the behavior of quadratic functions, which can be written as \(y = ax^2 + bx + c\). The graph of a quadratic function is a parabola, a symmetrical curve that can open upwards or downwards depending on the sign of \(a\).
When graphing the quadratic function \(y = x^2 + 3x - 5\), we begin by identifying the roots of its corresponding equation \(x^2 + 3x - 5 = 0\). Those roots, as found by the quadratic formula, are the x-intercepts of the parabola on the graph. For our example, the parabola will intersect the x-axis at \(x = 1\) and \(x = -5\).
By plotting these intercepts and the vertex, which is the highest or lowest point on the parabola, you can sketch the basic shape of the curve. It's important to note that the vertex can be found using the formula \(x = -\frac{b}{2a}\), and by substituting this \(x\) value into the original function, you can find the y-coordinate of the vertex. Understanding these elements of graphing quadratic functions empowers learners to decode the comprehensive story that a simple equation can tell.
When graphing the quadratic function \(y = x^2 + 3x - 5\), we begin by identifying the roots of its corresponding equation \(x^2 + 3x - 5 = 0\). Those roots, as found by the quadratic formula, are the x-intercepts of the parabola on the graph. For our example, the parabola will intersect the x-axis at \(x = 1\) and \(x = -5\).
By plotting these intercepts and the vertex, which is the highest or lowest point on the parabola, you can sketch the basic shape of the curve. It's important to note that the vertex can be found using the formula \(x = -\frac{b}{2a}\), and by substituting this \(x\) value into the original function, you can find the y-coordinate of the vertex. Understanding these elements of graphing quadratic functions empowers learners to decode the comprehensive story that a simple equation can tell.
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