Problem 32 - Calculus Early Transcendentals: Pearson New International Edition

Question

plot the graphs of both equations on the same coordinate plane. Find and label the points of intersection of the two graphs. $$ \begin{array}{l} y=2 x+3 \\ y=-(x-1)^{2} \end{array} $$

Step-by-Step Solution

Answer

The graphs intersect at the points (2, 7) and (-2, -1).

Step 1: Understand the Equations

We have two equations: the first is a linear equation \( y = 2x + 3 \) and the second is a quadratic equation \( y = -(x-1)^2 \). The task is to plot these equations on the same coordinate plane.

Step 2: Graph the Linear Equation

To graph the linear equation \( y = 2x + 3 \), plot the y-intercept at (0, 3) and use the slope (which is 2) to find another point. For example, from (0, 3), rise 2 units and run 1 unit to the right to reach the second point (1, 5). Draw a line through these points.

Step 3: Graph the Quadratic Equation

For \( y = -(x-1)^2 \), determine the vertex of the parabola, which is at (1, 0). Since the parabola opens downwards (due to the negative sign), plot additional points on either side of the vertex, such as (0, -1) and (2, -1). Sketch the downward opening parabola.

Step 4: Determine Points of Intersection

Set \( 2x + 3 = -(x-1)^2 \) to find the points of intersection by solving the equation. First, expand \( -(x-1)^2 \) to get \( -(x^2 - 2x + 1) = -x^2 + 2x - 1 \). Then, equate: \( 2x + 3 = -x^2 + 2x - 1 \). Simplify to get \( x^2 = 4 \), giving solutions for \( x \) as \( x = 2 \) and \( x = -2 \).

Step 5: Find Corresponding y-values

Substitute \( x = 2 \) and \( x = -2 \) back into any of the original equations to find corresponding \( y \)-values. For \( x = 2 \), \( y = 2(2) + 3 = 7 \). For \( x = -2 \), \( y = 2(-2) + 3 = -1 \).

Step 6: Plot Points of Intersection

The points of intersection are (2, 7) and (-2, -1). Plot these points on the graph and label them as points of intersection.

Key Concepts

Linear Equations

A linear equation is an equation that forms a straight line when graphed. The simplest form of a linear equation is given by the equation of a line: \[ y = mx + b \] where:

• \( m \) is the slope of the line, which shows how steep the line is. A larger slope means the line is steeper.
• \( b \) is the y-intercept, which is the point where the line crosses the y-axis.

For example, in the equation \( y = 2x + 3 \):

• The slope \( m \) is 2, indicating that for every 1 unit the x-value increases, the y-value increases by 2 units.
• The y-intercept \( b \) is 3, meaning the line cuts the y-axis at the point (0, 3).

To graph the line, start at the y-intercept, and use the slope to find another point. Then draw a straight line through both points.

Quadratic Equations

Quadratic equations represent parabolas when graphed. The standard form is:\[ y = ax^2 + bx + c \]The graph is a U-shaped curve called a parabola.

• If \( a > 0 \), the parabola opens upwards.
• If \( a < 0 \), it opens downwards.

For the given equation \( y = -(x-1)^2 \):

• The vertex form is \( y = a(x-h)^2 + k \), where \( (h, k) \) is the vertex.
• The vertex of the parabola is (1, 0).
• The negative sign in front of \((x-1)^2\) indicates the parabola opens downwards.

To plot, start at the vertex and then find points symmetrically around the vertex to shape the parabola.

Coordinate Plane

The coordinate plane is a two-dimensional surface where we plot points, lines, curves, and other shapes. It consists of:

• A horizontal axis, known as the x-axis.
• A vertical axis, known as the y-axis.
• The point where the x-axis and y-axis intersect is called the origin, labeled as (0,0).

Each point on the coordinate plane is identified by an ordered pair \((x, y)\). The x-value tells you how far to move left or right from the origin, and the y-value tells you how far to move up or down. By accurately plotting the points and sketching curves or lines, you can represent mathematical relationships visually.

Points of Intersection

Points of intersection are where two graphs, such as lines or curves, meet on a coordinate plane. To find these, set the equations representing the graphs equal to each other and solve for the variable. For example, to find where the line \( y = 2x + 3 \) intersects the parabola \( y = -(x-1)^2 \), solve the equation:\[ 2x + 3 = -(x-1)^2 \]This solution gives the x-values at which the graphs intersect. Substitute these x-values back into either of the original equations to find the corresponding y-values. For this exercise:

• The x-values of intersection are 2 and -2.
• The corresponding y-values are 7 for \( x = 2 \) and -1 for \( x = -2 \).

Thus, the points of intersection are (2, 7) and (-2, -1), which can be plotted and labeled on the graph.