Problem 45 - Precalculus with Calculus Previews

Question

In Problems \(45-48\), find the \(x\) - and \(y\) -intercepts of the given parabola. \((y+4)^{2}=4(x+1)\)

Step-by-Step Solution

Answer

The x-intercept is (3,0); the y-intercepts are (0,-2) and (0,-6).

Step 1: Rewrite the Equation in a Familiar Form

The given equation is \((y+4)^2 = 4(x+1)\). This is in the form of a horizontal parabola. To identify the intercepts, we compare it with the standard form \((y-k)^2 = 4p(x-h)\), recognizing that \(h = -1\) and \(k = -4\).

Step 2: Find the x-intercept

The x-intercept occurs where \(y = 0\). Substitute \(y = 0\) into the equation: \[(0+4)^2 = 4(x+1).\]Simplify to \(16 = 4(x+1)\). Solving for \(x\) gives \(x = 3\). Thus, the x-intercept is \((3,0)\).

Step 3: Find the y-intercepts

The y-intercept occurs where \(x = 0\). Substitute \(x = 0\) into the equation:\[(y+4)^2 = 4(0+1).\]This simplifies to \((y+4)^2 = 4\). Taking the square root of both sides gives \(y+4 = 2\) or \(y+4 = -2\). Solving these, we find \(y = -2\) or \(y = -6\). Thus, the y-intercepts are \((0,-2)\) and \((0,-6)\).

Key Concepts

Intercepts

When looking at the equation of a parabola, the intercepts are key points where the graph crosses the axes. The parabola can have both an *x-intercept* and a *y-intercept*.

• The **x-intercept** is the point where the parabola crosses the x-axis. This occurs when y is equal to 0. By substituting y = 0 into the parabola's equation, we can solve for x to find the x-intercept.
• The **y-intercept** is the point where the parabola crosses the y-axis. This happens when x is equal to 0. By setting x = 0 and solving for y, we find the y-intercept.

In the example given, we calculated the x-intercept at the point (3, 0) and the y-intercepts at the points (0, -2) and (0, -6). These points indicate where the parabola touches or crosses the axes, giving insight into its position and path in the coordinate plane.

Horizontal Parabola

A horizontal parabola is distinct from the more common vertical parabola. Instead of opening upward or downward, it opens to the right or the left.

• If a parabola is open**right**, its equation will involve y squared. The format might look like \((y-k)^2 = 4p(x-h)\).
• If open **left**, a negative sign introduces adjustments.

In our example, the equation \((y+4)^2 = 4(x+1)\) describes a parabola that opens to the right. This is identified by the squared y-term on the left side of the equation, meaning that changes in y don’t influence how wide or narrow the parabola is. Instead, changes affect how quickly the parabola moves horizontally along the x-axis.

Standard Form of Parabola

The standard form of a parabola provides a versatile way to determine its key properties quickly:

• The equation of a **horizontal parabola** takes the form \((y-k)^2 = 4p(x-h)\).
• In this standard form, \( (h, k) \) represents the vertex, which is the turning point of the parabola.
• The parameter \( p \) defines the distance from the vertex to the focus of the parabola.
• The sign and value of \( p \) indicate the direction and extent of the opening of the parabola.

In the specific problem, rewriting the equation to \((y+4)^2=4(x+1)\) allowed us to easily identify the vertex at (-1, -4) and understand the properties such as whether it's a horizontal or vertical orientation. Knowing the standard form makes solving and graphing parabolas much more straightforward.