Question

Finding Equations of Tangent Lines Sketch the graphs of \(y=x^{2}\) and \(y=-x^{2}+6 x-5\) , and sketch the two lines that are tangent to both graphs. Find equations of these lines.

Short answer

The equations of the tangent lines to both graphs are \(y=3x-2.25\) and \(y=3x-3.25\).

Step-by-step solution

Find the Derivative

Find the derivative of the functions to get the slope of the tangent lines. The derivative of \(y=x^{2}\) is \(y'=2x\) and the derivative of \(y=-x^{2}+6x-5\) is \(y'=-2x+6\).

Set the Derivatives Equal

Since the tangent lines touch both curves, they have the same slope at the points of tangency. So, you need to solve \(2x=-2x+6\), which simplifies to \(4x=6\), resulting in \(x=1.5\).

Find the Points of Tangency

Substitute \(x=1.5\) into both equations to get the y-coordinates of the points of tangency. For \(y=x^{2}\), \(y=(1.5)^{2}=2.25\). For \(y=-x^{2}+6x-5\), \(y=-1.5^{2} +6(1.5) -5 = 1.25\). The points of tangency are \((1.5, 2.25)\) and \((1.5, 1.25)\).

Find the Equations of the Tangent Lines

Use the point-slope form of a line equation to find the equations. The tangent line to \(y=x^{2}\) is \(y -2.25 = 3(x - 1.5)\), or \(y=3x-2.25\). The tangent line to \(y=-x^{2}+6x-5\) is \(y -1.25 = 3(x - 1.5)\), or \(y=3x-3.25\). You have found the equations of the two lines that are tangent to both graphs.

Key concepts

Derivatives

Derivatives are a fundamental tool in calculus that measure the rate at which a function changes as its input changes. They are essentially the slope of the tangent line to a curve at a specific point. Understanding derivatives is essential when finding the equation of a tangent line, because the derivative gives us the slope at any given point on the function.

When dealing with simple power functions like y = x^2, finding the derivative is straightforward. For y = x^2, the derivative, denoted as y', is 2x. This means that for any x-value, multiplying it by 2 will give the slope of the tangent line at that point. Similarly, for a more complex function like y = -x^2 + 6x - 5, applying the power rule and the constant rule for each term gives us y' = -2x + 6 as the derivative.

Derivatives not only determine the slope but also help us identify where functions are increasing or decreasing, which is pivotal in sketching the direction of tangent lines.

Tangency Points

Tangency points are the specific points on a curve where a tangent line touches it. These points are critical because the tangent line only intersects the curve at that solitary point and does not cut across the curve. In order to find the tangency points, one must first determine the slope of the tangent line using derivatives, as mentioned in the previous section.

After determining the slope, we can set the slopes of the derivatives of the two functions equal to each other, as the tangent lines to both will share the same slope at the points of tangency. Solving the equation generated provides the x-value at which tangent occurs. Substituting this x-value back into the original functions gives us the y-coordinates, thus identifying the actual points on the curve where the tangent line will touch. These points serve as vital parts of the equation when determining the final form of the tangent line.

Slope of Tangent Line

The slope of the tangent line represents how steep or flat the line is at the point it touches the curve, which is visually the angle at which the line intersects the Cartesian plane. As stated, to find the slope of the tangent line, we need to find the derivative of the function at the specific point of tangency.

In our example, after calculating the derivatives of y = x^2 and y = -x^2 + 6x - 5, we determined that at the point of tangency, the slopes of the tangent lines must be the same. By setting the derivatives equal to each other, 2x = -2x + 6, we found the x-value that gives the slope of the tangent lines when they touch both curves. Thus, we solved for the common slope of the tangent lines at x = 1.5, which turned out to be 3. This value is crucial as it is used in the point-slope form of the equation to derive the actual equations of the tangent lines.

Point-Slope Form

The point-slope form is a method for writing the equation of a line when you know the slope and a single point on the line (in this case, the tangency point). The general formula is (y - y1) = m(x - x1), where (x1, y1) is the point on the line, and m is the slope.

With the slope determined from the derivative and the points of tangency calculated, using the point-slope form is straightforward. For our example with the points (1.5, 2.25) and (1.5, 1.25) on the respective curves and a common slope of 3, the equations of the tangent lines are derived by substituting these values into the point-slope formula. This results in y - 2.25 = 3(x - 1.5) for the first line and y - 1.25 = 3(x - 1.5) for the second, which can be simplified to obtain the final equations of the tangent lines. This form is incredibly useful because it provides a quick way to find the equation of a line in a context where the slope and a specific point are known.