Problem 53 - Precalculus : Building Concepts and Connections

Question

Find the vertex and axis of symmetry of the associated parabola for each quadratic function. Sketch the parabola. Find the intervals on which the function is increasing and decreasing, and find the range. $$f(x)=2 x^{2}+4 x-3$$

Step-by-Step Solution

Answer

The vertex of the function is \((-1, -1)\), the axis of symmetry is \(x=-1\), the function is decreasing in the interval \(-∞ < x < -1\), increasing in the interval \(-1 < x < ∞\), and has a range of \([-1, ∞)\).

Step 1: Find the vertex

The vertex, \((h, k)\), of a quadratic function given as \(f(x) = a(x-h)^{2} + k\), can be found using the formulas: \(h = -\frac{b}{2a}\) and \(k = f(h)\). For the given function \(f(x) = 2x^{2} + 4x - 3\), \(a = 2\) and \(b = 4\). Plugging these into the formula gives \(h = -\frac{4}{2*2} = -1\). The y-coordinate of the vertex (\(k\)) can be found by plugging \(h\) into the function, \(k = f(-1)\), which gives \(k = 2(-1)^{2} + 4(-1) - 3 = -1\). So, the vertex of the function is \((-1, -1)\).

Step 2: Identify the axis of symmetry

The axis of symmetry of a parabola is always the x-coordinate of the vertex (h). Hence, for this function, the axis of symmetry is \(x = -1\).

Step 3: Plot the graph

Having the vertex, \((-1,-1)\), and the axis of symmetry, \(x=-1\), along with the knowledge that the coefficient of \(x^{2}\) is positive – meaning the parabola opens upward – allows for an accurate plot of the parabola.

Step 4: Determine the intervals of increase and decrease

Since the parabola opens upwards, the function decreases before the vertex and increases after. Therefore, \(f(x)\) is decreasing in the interval \(-∞ < x < -1\), and is increasing in the interval \(-1 < x < ∞\).

Step 5: Find the range of the function

From the vertex, it can be seen that the minimum value of \(y\) is \(-1\), and since the parabola opens upward, the maximum is \(∞\). Therefore, the range of the function is \([-1, ∞)\).

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a key point that helps to describe its shape and position on the graph. For any quadratic function in the form of \( f(x) = ax^2 + bx + c \), the vertex can be found using a simple formula.

Here's how you can do it:

• The x-coordinate (\( h \)) of the vertex is given by \( h = -\frac{b}{2a} \).
• The y-coordinate (\( k \)) is calculated by plugging this \( h \) value into the function \( f(h) \).

For the quadratic function \( f(x) = 2x^2 + 4x - 3 \):

• The value \( a \) is 2, and \( b \) is 4.
• Substitute these values into \( h = -\frac{b}{2a} \) to find \( h = -1 \).
• Substitute \( h = -1 \) into the function to find \( k = 2(-1)^2 + 4(-1) - 3 = -1 \).

Thus, the vertex of the parabola described by this quadratic function is \((-1, -1)\). This point marks the minimum or the maximum point of the parabola, depending on the direction it opens.

Axis of Symmetry

The axis of symmetry is an imaginary line that perfectly divides the parabola into two symmetrical halves. It's an essential characteristic of parabolas and aids in graphing.
For a quadratic function, this line is vertical and corresponds to the x-coordinate of the vertex of the parabola.

Using the earlier example of the quadratic function \( f(x) = 2x^2 + 4x - 3 \):

• The vertex was found to be at \((-1, -1)\).
• Therefore, the axis of symmetry is the vertical line \( x = -1 \).

This line tells us that if you were to "fold" the parabola along \( x = -1 \), both sides of the parabola would align perfectly. It also hints that for every point \((x, y)\) on the left side of the axis, there is a corresponding point \((x', y)\) right of the axis, maintaining symmetry.

Intervals of Increase and Decrease

Understanding intervals of increase and decrease helps in understanding how a function behaves over different parts of its domain.

For a parabolic curve described by a quadratic function, the behavior of the parabola in these intervals is directly related to its vertex:

• If the parabola opens upward (as in \( f(x) = 2x^2 + 4x - 3 \)), it means the curve decreases to the left of the vertex and increases to the right.
• The opposite is true if the parabola opens downward.

For our function:

• The vertex is at \( x = -1 \), thus:
• The function decreases for \( x < -1 \), i.e., in the interval \((-\infty, -1)\).
• The function increases for \( x > -1 \), which is \((-1, \infty)\).

Being able to identify these intervals is crucial for understanding the nature of the changes in the values of the function and in practical applications like graphing.

Range of a Function

The range of a function is all the possible y-values it can produce, essentially capturing the height of the graph on the coordinate system.

In quadratic functions, the range is determined by:

• Whether the parabola opens upward or downward.
• The y-coordinate of the vertex, which serves as the minimum or maximum point.

For the given function \( f(x) = 2x^2 + 4x - 3 \):

• The parabola opens upward because the coefficient of \( x^2 \) is positive.
• The vertex is \( (-1, -1) \), thus the lowest point on the parabola is \( y = -1 \).

From this information, the range can be determined as being all y-values from \(-1\) to \( \infty \), or simply \([-1, \infty)\). This indicates that \( f(x) \) can take on any y-value greater than or equal to \(-1\), extending upwards indefinitely.