Problem 47 - Precalculus with Calculus Previews

Question

Find the \(x\) - and \(y\) -intercepts of the given parabola. \(x^{2}+2 y-18=0\)

Step-by-Step Solution

Answer

The x-intercepts are \( x = 3\sqrt{2} \) and \( x = -3\sqrt{2} \); the y-intercept is \( y = 9 \).

Step 1: Set the formula for finding x-intercepts

To find the x-intercepts, set the equation equal to zero for y. In general form, the x-intercepts occur where \(y = 0\). Substitute \(y = 0\) into the equation: \(x^2 + 2(0) - 18 = 0\). This simplifies to \(x^2 - 18 = 0\).

Step 2: Solve for x-intercepts

Solve the equation \(x^2 - 18 = 0\) by isolating \(x^2\):\[ x^2 = 18 \]Now, take the square root of both sides:\[ x = \pm \sqrt{18} \]This simplifies to \( x = \pm 3\sqrt{2}\). Therefore, the x-intercepts are \( x = 3\sqrt{2} \) and \( x = -3\sqrt{2} \).

Step 3: Set the formula for finding y-intercepts

To find the y-intercepts, set the equation equal to zero for x. In general form, the y-intercepts occur where \(x = 0\). Substitute \(x = 0\) into the equation: \(0^2 + 2y - 18 = 0\). This simplifies to \(2y - 18 = 0\).

Step 4: Solve for y-intercepts

Solve the equation \(2y - 18 = 0\) by isolating \(y\):\[ 2y = 18 \]Divide both sides by 2:\[ y = 9 \]Thus, the y-intercept is \( y = 9 \).

Key Concepts

x-intercepts

When dealing with a parabola, finding the \(x\)-intercepts is essential for understanding where the graph crosses the \(x\)-axis. Simply put, these are the points on the graph where the \(y\) value is zero. For instance, in the equation \(x^{2} + 2y - 18 = 0\), we find the \(x\)-intercepts by setting \(y = 0\). This transforms the equation into \(x^2 - 18 = 0\).
To solve this, we isolate \(x^2\), resulting in \(x^2 = 18\). Taking the square root of both sides, we obtain \(x = \pm \sqrt{18}\), which simplifies further to \(x = \pm 3\sqrt{2}\). Therefore, the \(x\)-intercepts are \(x = 3\sqrt{2}\) and \(x = -3\sqrt{2}\), indicating that the parabola crosses the \(x\)-axis at these values.

y-intercepts

The \(y\)-intercept is where the graph of an equation crosses the \(y\)-axis. It signifies the point at which \(x = 0\). For the given parabola equation, \(x^{2} + 2y - 18 = 0\), this means setting \(x = 0\).
Thus, the equation simplifies to \(2y - 18 = 0\). To find the value of \(y\), solve for \(y\) by isolating it on one side of the equation. By adding 18 to both sides, we get \(2y = 18\).
Further simplification by dividing both sides by 2 results in \(y = 9\). Hence, the parabola intersects the \(y\)-axis at \(y = 9\). This point is significant as it helps in graphing the parabola accurately.

solving quadratic equations

Quadratic equations are expressions of the form \(ax^2 + bx + c = 0\). Solving these equations involves finding the values of \(x\) that satisfy the equation. Methods to solve quadratic equations include factoring, using the quadratic formula, completing the square, and more.
In our case for \(x^2 - 18 = 0\), we utilized the method of taking the square root. Solving by square roots is especially useful when the equation is set in the format \(x^2 = k\). To solve, simply isolate \(x^2\), and then apply the square root to both sides.
This method revealed the solutions \(x = \pm 3\sqrt{2}\). It's a straightforward approach when compared to other methods like factoring or using the quadratic formula, which might be necessary in more complex cases.