Problem 55 - A Graphical Approach to College Algebra

Question

Draw a sketch of the two graphs described with the indicated number of points of intersection. (There may be more than one way to do this.) A line and a parabola; one point.

Step-by-Step Solution

Answer

Sketch a parabola \(y = x^2\) and tangent line \(y = 2x - 1\) meeting at (1, 1).

Step 1: Understand the Graphs Involved

A parabola is the graph of a quadratic function, typically written as \(y = ax^2 + bx + c\). A line is the graph of a linear function, typically expressed as \(y = mx + b\). We need to create graphs for these so that they intersect at exactly one point.

Step 2: Determine Intersection Conditions

The line must be tangent to the parabola so that they intersect at exactly one point. This means the line will just touch the parabola without crossing it, resulting in a single intersection point.

Step 3: Choose a Parabola

For simplicity, let's choose the parabola \(y = x^2\), which is centered at the origin and opens upwards.

Step 4: Find a Tangent Line

To find a line that is tangent to \(y = x^2\), we need it to have the same slope at the point of tangency. This can be found by deriving the parabola: the derivative \(y' = 2x\) gives the slope of the tangent line at any point \(x\). Choose a point, say \(x = 1\), then \(y = 1\), and the slope is 2.

Step 5: Equation of Tangent Line

The tangent line at the point \((1, 1)\) with slope 2 has the equation \(y - 1 = 2(x - 1)\). Simplifying, this becomes \(y = 2x - 1\).

Step 6: Verify Intersection

Set \(x^2 = 2x - 1\) to find intersection points. Solving \(x^2 - 2x + 1 = 0\), we get \((x - 1)^2 = 0\), so \(x = 1\). At \(x = 1\), \(y = 1\), confirming the line and parabola intersect only at \((1, 1)\).

Step 7: Sketch the Graphs

On a coordinate plane, sketch the parabola \(y = x^2\) and the line \(y = 2x - 1\). Ensure the line just touches the parabola at the point \((1,1)\), showing one intersection.

Key Concepts

Quadratic Functions

Quadratic functions are a key concept in algebra, often represented by the equation \(y = ax^2 + bx + c\). This equation forms a parabolic shape when graphed on a coordinate plane. The parabola can open upwards or downwards depending on the sign of the coefficient \(a\). Here, we examine the simplest form, \(y = x^2\), which opens upwards with its vertex at the origin (0,0).

The graph of a quadratic function can be characterized by:

• Vertex: The highest or lowest point on the graph (turning point).
• Axis of symmetry: A vertical line that runs through the vertex, dividing the parabola into two symmetrical halves.
• Direction of opening: Determined by the coefficient of \(x^2\). Positive for upwards and negative for downwards.

Understanding these characteristics is crucial when graphing or analyzing the parabola, especially when determining intersections with other functions, like linear functions.

Linear Functions

Linear functions are perhaps the simplest type of mathematical function. They are expressed in the form \(y = mx + b\), where \(m\) represents the slope and \(b\) is the y-intercept. The graph of a linear function is a straight line.

Here are some key points:

• Slope (m): Determines the steepness and direction of the line. A positive slope means the line ascends from left to right, while a negative slope descends.
• Y-intercept (b): The point where the line crosses the y-axis.
• Intercepts: Besides the y-intercept, the x-intercept, where the line crosses the x-axis, is also crucial in forming a complete graph.

In the context of graphing intersections, such as the exercise with the parabola \(y = x^2\), it is important to determine the line's equation so it intersects at the desired point, for example, \((1,1)\) as the tangent.

Tangent Lines

Tangent lines play a critical role in calculus and geometry, especially when discussing intersections with curves. A tangent line is a straight line that touches a curve at a single point, known as the point of tangency, without crossing it. This implies that at this point, the line and the curve have the same slope.

In the case of a parabola like \(y = x^2\), we can find the tangent by employing calculus:

• Derivative: The derivative \(y' = 2x\) gives us the slope of the tangent line at any point on the parabola.
• Point of Tangency: To find the tangent at a specific point, such as \((1, 1)\), substitute \(x = 1\) into the derivative to find the slope, which is 2.
• Tangent Line Equation: Use the point-slope form \(y - y_1 = m(x - x_1)\), where \(m = 2\) and \((x_1, y_1) = (1, 1)\), resulting in the equation \(y = 2x - 1\).

These tangent lines are useful for precise analysis and graphing tasks, such as showing the exact point of intersection with another graph.