Problem 18 - A Graphical Approach to Precalculus with Limits

Question

Match each function in Column I with the description of the parabola that is its graph in Column II, assuming \(a>0, h>0,\) and \(k>0\). \(\mathbf{I}\) (a) \(f(x)=-a(x+h)^{2}+k\) (b) \(f(x)=a(x-h)^{2}+k\) (c) \(f(x)=a(x+h)^{2}+k\) (d) \(f(x)=-a(x-h)^{2}+k\) \(\mathbf{II}\) A. Vertex in quadrant I, two \(x\)-intercepts B. Vertex in quadrant I, no \(x\)-intercepts C. Vertex in quadrant II, two \(x\)-intercepts D. Vertex in quadrant II, no \(x\)-intercepts

Step-by-Step Solution

Answer

a-D, b-B, c-C, d-A.

Step 1: Analyze the vertex form of a parabola

The function \(f(x) = a(x-h)^2 + k\) describes a parabola with vertex at \((h, k)\). If \(a > 0\), the parabola opens upwards, and if \(a < 0\), it opens downwards. The sign of \(h\) and \(k\) dictates in which quadrant the vertex lies.

Step 2: Match function (a) with description

For \(f(x)=-a(x+h)^{2}+k\), since \(-a < 0\), the parabola opens downwards. The vertex is at \((-h, k)\). Here, \(-h < 0\) and \(k > 0\), so the vertex is in quadrant II. Since it opens downwards, it has no \(x\)-intercepts. Thus, it matches description D.

Step 3: Match function (b) with description

For \(f(x)=a(x-h)^{2}+k\), \(a > 0\) means the parabola opens upwards. The vertex is \((h, k)\). Since \(h > 0\) and \(k > 0\), the vertex is in quadrant I. This parabola has no \(x\)-intercepts because it opens upwards above the \(x\)-axis. Thus, it matches description B.

Step 4: Match function (c) with description

For \(f(x)=a(x+h)^{2}+k\), \(a > 0\) means the parabola opens upwards. The vertex is \((-h, k)\), placing it in quadrant II because \(-h < 0\) and \(k > 0\). With vertex in quadrant II and opening upwards, the parabola has two \(x\)-intercepts. Thus, it matches description C.

Step 5: Match function (d) with description

For \(f(x)=-a(x-h)^{2}+k\), \(-a < 0\) means the parabola opens downwards. The vertex is \((h, k)\). Given \(h > 0\) and \(k > 0\), the vertex is in quadrant I. It opens downwards so it crosses the \(x\)-axis twice. Thus, it matches description A.

Key Concepts

Vertex Form

To describe the shape and position of a parabola on a graph, the vertex form equation is essential. This specific form is given by the equation \(f(x) = a(x-h)^2 + k\). The vertex of the parabola, which is its highest or lowest point depending on the opening, is located at the coordinates \((h, k)\). The value of \(a\) determines the direction in which the parabola opens:

• If \(a > 0\), the parabola opens upwards.
• If \(a < 0\), the parabola opens downwards.

Knowing the vertex form allows us to easily identify these features, helping us understand the parabola’s behavior and position.

Quadrants

When considering the position of the parabola's vertex, it is crucial to understand the four quadrants of the Cartesian plane. Quadrants are determined based on the signs of the \(h\) and \(k\) values in the vertex form:

• Quadrant I: \((h, k)\) where \(h > 0\) and \(k > 0\).
• Quadrant II: \((h, k)\) where \(h < 0\) and \(k > 0\).
• Quadrant III and IV are not considered here since the exercise focuses on the first two quadrants.

Positioning the vertex in the correct quadrant is essential for predicting the general graph shape and understanding where it resides on the coordinate plane.

X-intercepts

The \(x\)-intercepts are the points where the parabola crosses the x-axis. They are also known as the roots or zeros of a quadratic function, typically determined by setting \(f(x) = 0\). Depending on how a parabola is positioned relative to the x-axis, it may have:

• Two x-intercepts: when the parabola crosses the x-axis twice.
• No x-intercepts: if the parabola does not intersect the x-axis at all.

These intercepts provide critical insight into the solutions of the quadratic equation represented by the parabola, indicating whether it consistently stays above or below the x-axis.

Upwards and Downwards Opening

A principal characteristic of a parabola is whether it opens upwards or downwards. This is determined by the sign of the coefficient \(a\):

• An upwards opening (\(a > 0\)) means the parabola is shaped like the letter 'U', with the vertex being the lowest point.
• A downwards opening (\(a < 0\)) implies the parabola is like an upside-down 'U', with the vertex at its peak.

Understanding the direction in which a parabola opens is crucial for interpreting its behavior in terms of minimum and maximum values, as well as its range and intercepts. This helps predict how the parabola interacts with the axes and surrounding space.