Problem 48 - Precalculus with Calculus Previews

Question

Find the \(x\) - and \(y\) -intercepts of the given parabola. \(y^{2}-8 y-x+15=0\)

Step-by-Step Solution

Answer

The x-intercept is (15, 0) and the y-intercepts are (0, 3) and (0, 5).

Step 1: Express the equation in terms of x

To find the intercepts, we need to express the equation in terms of either \( x \) or \( y \). For the \( y \)-intercept, let's express in terms of \( x \): Start by isolating \( x \): \[\begin{align*} y^2 - 8y + 15 &= x\end{align*}\]

Step 2: Calculate the y-intercept

The \( y \)-intercept occurs where \( x = 0 \). Substitute \( x = 0 \) into the equation: \[ y^2 - 8y + 15 = 0 \] Factor the equation: \[ (y-3)(y-5) = 0\] Thus, the \( y \)-intercepts are at \( y = 3 \) and \( y = 5 \). So, the coordinates for the \( y \)-intercepts are \((0, 3)\) and \((0, 5)\).

Step 3: Calculate the x-intercept

The \( x \)-intercept occurs when \( y=0 \). Substitute \( y = 0 \) into the original equation:\[ 0^2 - 8(0) - x + 15 = 0 \] Solve for \( x \):\[ -x + 15 = 0 \] \[ x = 15 \] Thus, the \( x \)-intercept is \( x = 15 \), and the coordinate is \((15, 0)\).

Key Concepts

Understanding the y-intercept

The y-intercept of a parabola is the point where the graph of the equation crosses the y-axis. This is a key feature in the graphing of quadratic relations. It occurs at the point where the value of x is zero. For any given quadratic equation, setting x to zero allows us to solve for y, determining where the parabola intersect the y-axis.

To find the y-intercept, substitute x with 0 in the equation and solve for y. For instance, in the exercise provided, the equation is given by:

• \( y^2 - 8y + 15 = x \)

Setting \( x = 0 \) allows us to simplify the equation to:

• \( y^2 - 8y + 15 = 0 \)

Factoring this quadratic equation results in:

• \( (y-3)(y-5) = 0 \)

This gives us the y-intercepts at points \( y = 3 \) and \( y = 5 \). Therefore, the coordinates for the y-intercepts are \((0, 3)\) and \((0, 5)\).

Identifying the y-intercept is crucial as it provides valuable insight into the positioning of the parabola on a Cartesian coordinate system, enhancing one's understanding of the quadratic's behavior at the y-axis.

Understanding the x-intercept

The x-intercept of a parabola is the point where the graph crosses the x-axis. This occurs when the value of y is zero. Finding the x-intercept helps in understanding how the parabola interacts with the horizontal axis.

To determine the x-intercept, you need to set y to 0 in the quadratic equation and solve for x. For the example equation:

• \( y^2 - 8y - x + 15 = 0 \)

Substitute \( y = 0 \) to simplify the expression:

• \(-x + 15 = 0 \)

Solving for x gives us:

• \( x = 15 \)

Thus, the x-intercept is at the point \((15, 0)\).

The x-intercept is a pivotal component in graphing because it shows where the parabola intersects the x-axis, offering an important reference point for the shape and orientation of the graph.

Exploring coordinate geometry concepts

Coordinate geometry, also known as analytic geometry, combines algebra and geometry to study shapes, sizes, and positions of figures using coordinate points. It provides a powerful tool for graphing and visualizing complex equations like parabolas.

At the heart of coordinate geometry is the Cartesian plane, a grid defined by two perpendicular axes, x and y, which intersect at the origin point \((0, 0)\). Learning to navigate this grid allows one to place points, lines, and curves, effectively visualizing equations like the quadratic provided in the exercise:

• \( y^2 - 8y - x + 15 = 0 \)

Understanding how to find intercepts is fundamental in coordinate geometry because:

• The y-intercept shows where a graph crosses the y-axis, giving insight into its upward or downward shift.
• The x-intercept reveals where a graph intersects the x-axis, indicating zeros or roots of the function.

Mastering coordinate geometry in the context of quadratic equations equips students with essential graphing skills and the ability to predict the behavior of parabolas. Thus enhancing their problem-solving strategies in mathematics.