Problem 52 - Algebra A Combined Function

Question

Find the vertex of the graph of each quadratic function. Determine whether the graph opens upward or downward, find the y-intercept, approximate the \(x\) -intercepts to one decimal place, and graph the function. $$ f(x)=2 x^{2}+4 x-1 $$

Step-by-Step Solution

Answer

Vertex: (-1, -3); Opens upward; Y-intercept: (0, -1); X-intercepts: ~ (0.2, -2.2).

Step 1: Determine the Direction of the Parabola

The given quadratic function is in the form \( f(x) = ax^2 + bx + c \). Here, \( a = 2 \), \( b = 4 \), and \( c = -1 \). The parabola opens **upwards** because \( a > 0 \).

Step 2: Find the Vertex

The vertex \((h, k)\) of a parabola given by the quadratic equation \( ax^2 + bx + c \) can be found using the formulas: \( h = -\frac{b}{2a} \) and \( k = f(h) \).Calculate \( h \):\[ h = -\frac{4}{2(2)} = -\frac{4}{4} = -1\]Substitute \( x = -1 \) into the function:\[ k = 2(-1)^2 + 4(-1) - 1 = 2 - 4 - 1 = -3\]Thus, the vertex is \((-1, -3)\).

Step 3: Find the Y-intercept

The y-intercept occurs when \( x = 0 \). Substitute \( x = 0 \) into the function:\[ f(0) = 2(0)^2 + 4(0) - 1 = -1\]Thus, the y-intercept is \((0, -1)\).

Step 4: Find the X-intercepts

To find the x-intercepts, set \( f(x) = 0 \):\[ 2x^2 + 4x - 1 = 0\]Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here, \( a = 2 \), \( b = 4 \), \( c = -1 \):\[ x = \frac{-4 \pm \sqrt{4^2 - 4 \cdot 2 \cdot (-1)}}{2 \cdot 2} = \frac{-4 \pm \sqrt{16 + 8}}{4} = \frac{-4 \pm \sqrt{24}}{4}\]Approximate \( \sqrt{24} \approx 4.9 \):\[ x = \frac{-4 \pm 4.9}{4}\]Thus, the x-intercepts are approximately:\[ x \approx \frac{-4 + 4.9}{4} = 0.225\quad \text{and}\quad x \approx \frac{-4 - 4.9}{4} = -2.225\]

Step 5: Graph the Function

Plot the vertex \((-1, -3)\), the y-intercept \((0, -1)\), and the x-intercepts approximately \((0.2, 0)\) and \((-2.2, 0)\). Connect these points with a smooth U-shaped curve opening upwards, reflecting the nature of the quadratic function with \( a > 0 \).

Key Concepts

Vertex Calculation

The vertex of a quadratic function is a crucial point, as it shows the peak or the lowest point of the parabola, depending on its direction. For a quadratic equation in the standard form \( ax^2 + bx + c \), the vertex can be directly calculated using two specific formulas for its coordinates, \( h \) and \( k \).
The horizontal location of the vertex is given by:

• \( h = -\frac{b}{2a} \).

Once you have found \( h \), you can determine the vertical position \( k \) by plugging \( h \) back into the original function \( f(x) \):

• \( k = f(h) \).

In this function, \( f(x) = 2x^2 + 4x - 1 \), we calculate \( h = -1 \) and \( k = -3 \), making the vertex \((-1, -3)\). This point tells you where the parabola reaches its minimum because the parabola opens upwards. The vertex not only marks the minimum value but is also the point around which the parabola is symmetric.

Parabola Direction

Understanding the direction in which a parabola opens is central to sketching or analyzing quadratic functions. The direction is determined by the coefficient \( a \) in the quadratic equation \( f(x) = ax^2 + bx + c \).
If \( a > 0 \), the parabola opens **upwards**, creating a 'U' shape. This indicates that the vertex is the lowest point, or the minimum, on the graph.
Conversely, if \( a < 0 \), the parabola opens **downwards** resembling an '∩' shape, where the vertex is the peak or maximum.
In our exercise, since \( a = 2 \), which is positive, the graph of the function \( f(x) = 2x^2 + 4x - 1 \) clearly opens upwards. Understanding this aspect helps determine not only the direction but also the overall nature of the parabola's symmetry and range.

X-intercepts

The x-intercepts of a parabola are points where the graph crosses the x-axis. They are found by setting the function equal to zero and solving for \( x \).
This means solving the equation:

• \( 2x^2 + 4x - 1 = 0 \).

To find these points, you can use the quadratic formula:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

Substituting \( a = 2 \), \( b = 4 \), and \( c = -1 \) into the formula provides us with the x-intercepts approximately \( 0.225 \) and \( -2.225 \).
These intercepts indicate where the parabola crosses the x-axis. Knowing the x-intercepts is valuable for graphing and understanding the real solutions of the equation as they represent the roots of the function. The intercepts provide essential points to plot when sketching the graph, giving structure to the shape of the parabola.