Problem 31
Question
Solving a System of Equations Graphically in Exercises \(33-44,\) solve the system graphically. $$\left\\{\begin{array}{l}{x-y=-1} \\ {x^{2}-y=-4}\end{array}\right.$$
Step-by-Step Solution
Verified Answer
The solution to the system of equations is approximately where the line and the parabola intersect on the graph. The exact values of where the line and the parabola intersect depends on the precision of your graphing.
1Step 1: Graphing the first equation
Rewrite the first equation in the form \(y = x + 1\). This is a linear equation with slope 1 and y-intercept 1. To graph this equation, start at the point (0,1) on the y-axis (the y-intercept) and use the slope to find other points. The slope of 1 indicates to go up one unit and right one unit from the y-intercept. Continue this until you have enough points to draw a straight line.
2Step 2: Graphing the second equation
Rewrite the second equation in the form \(y = x^{2} - 4\). This is a parabolic equation with vertex at (0,-4). To graph this equation, start at the point (0,-4) on the y-axis (the vertex of the parabola), then plot some points above and below it. Observing the equation, we see that the curve opens upwards. Continue plotting points until you have enough to draw a smooth curve.
3Step 3: Finding the points of intersection
Observe where the line from the first equation intersects (crosses) the curve from the second equation. These points of intersection are the solutions to the system of equations. If necessary, you may have to estimate the coordinates of the points of intersection.
Key Concepts
Graphing Linear EquationsGraphing ParabolasPoints of IntersectionQuadratic Functions
Graphing Linear Equations
To visualize a linear equation like the one in our exercise, \(y = x + 1\), we start by identifying two crucial components: the slope and the y-intercept. The slope is the rate at which y increases as x increases. Here, the slope is 1, which means for each step we move to the right along the x-axis, the y value increases by the same amount.
Graphing a linear equation is essentially connecting the dots. After plotting the y-intercept at \(0, 1\), we use the slope to find other points. For example, from the point \(0, 1\), move one unit up and one unit right to reach the next point \(1, 2\). Repeating this process gives us a line of evenly spaced dots that we connect with a straight edge. The line drawn represents all possible solutions to the equation \(y = x + 1\).
Graphing a linear equation is essentially connecting the dots. After plotting the y-intercept at \(0, 1\), we use the slope to find other points. For example, from the point \(0, 1\), move one unit up and one unit right to reach the next point \(1, 2\). Repeating this process gives us a line of evenly spaced dots that we connect with a straight edge. The line drawn represents all possible solutions to the equation \(y = x + 1\).
Graphing Parabolas
Graphing the equation \(y = x^{2} - 4\) helps us bring the shape of a parabola to life. A parabola is a symmetrical, U-shaped curve. In this equation, the vertex, or the lowest point of the parabola, is at \(0, -4\). The vertex serves as a starting point for graphing.
To graph this parabola, we plot the vertex and then find additional points on either side of it. For every positive and negative x-value, compute the corresponding y-value and plot the points. Since the coefficient of \(x^{2}\) is positive, the parabola opens upwards. As we move away from the vertex, the y-values increase rapidly, forming the distinctive U-shape. Connecting these points creates a smooth curve, illustrating all the sets of \(x, y\) that satisfy the quadratic equation.
To graph this parabola, we plot the vertex and then find additional points on either side of it. For every positive and negative x-value, compute the corresponding y-value and plot the points. Since the coefficient of \(x^{2}\) is positive, the parabola opens upwards. As we move away from the vertex, the y-values increase rapidly, forming the distinctive U-shape. Connecting these points creates a smooth curve, illustrating all the sets of \(x, y\) that satisfy the quadratic equation.
Points of Intersection
The points of intersection are where the graphs of two equations cross each other. These are significant because they represent the pairs of \(x, y\) that are solutions to both equations in the system. When we graphed our linear equation and our parabola, we looked for where their paths meet.
On paper, finding the exact points of intersection may require a little estimation, unless the intersection occurs at whole number coordinates. In some cases, we might use grid lines to help us approximate these points. In more advanced work, algebraic methods or technology tools can provide precise coordinates. After identifying the points of intersection, it's critical to verify that the coordinates satisfy both original equations.
On paper, finding the exact points of intersection may require a little estimation, unless the intersection occurs at whole number coordinates. In some cases, we might use grid lines to help us approximate these points. In more advanced work, algebraic methods or technology tools can provide precise coordinates. After identifying the points of intersection, it's critical to verify that the coordinates satisfy both original equations.
Quadratic Functions
Quadratic functions are a type of polynomial represented generally as \(y = ax^{2} + bx + c\), where \(a, b, c\) are constants, and \(a \eq 0\). The graph of a quadratic function is a parabola, and its most basic form, \(y = x^{2}\), provides a symmetrical curve about the y-axis.
The key features of a parabola include the vertex, the axis of symmetry, the direction in which it opens (upwards or downwards), and its width (determined by the value of \(a\)). When \(a > 0\), the parabola opens up, and when \(a < 0\), it opens down. The larger the absolute value of \(a\), the narrower the parabola. Quadratic functions model many real-world scenarios, such as the path of a thrown ball and are fundamental to algebra and beyond.
The key features of a parabola include the vertex, the axis of symmetry, the direction in which it opens (upwards or downwards), and its width (determined by the value of \(a\)). When \(a > 0\), the parabola opens up, and when \(a < 0\), it opens down. The larger the absolute value of \(a\), the narrower the parabola. Quadratic functions model many real-world scenarios, such as the path of a thrown ball and are fundamental to algebra and beyond.
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