Problem 35 - Algebra and Trigonometry

Question

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. $$ f(x)=2 x^{2}+4 x-3 $$

Step-by-Step Solution

Answer

The vertex of the parabola is \((-0.5, -3.5)\), the x-intercepts can be found by solving \(2x^{2}+4x-3=0\) and the y-intercept is \(-3\). The equation of the axis of symmetry is \(x=-0.5\). The domain of the function is \(-\infty, \infty\) and the range is \([-3.5, \infty)\].

Step 1: Finding the Vertex

The vertex formula for a quadratic function \(f(x)=a(x-h)^{2}+k\) is given by \((h,k)\), where \(h=-\frac{b}{2a}\) and \(k=f(h)\). For this quadratic function, \(a=2\) and \(b=4\), substituting these values into the vertex formula, the vertex, \(h\), can be found as \(h=-\frac{b}{2a}=-\frac{4}{2*2}=-0.5\). Substituting \(h=-0.5\) into the function, \(k=2*(-0.5)^{2}+4*(-0.5)-3=-3.5\). So, the vertex is \((-0.5, -3.5)\).

Step 2: Calculating the Intercepts

To find the x-intercepts, set \(f(x)=2x^{2}+4x-3=0\). Solving this equation will produce the x-intercepts. Similarly, the y-intercept is found by substituting \(x=0\) into the equation, and solving for \(y\). This will give \(f(0)=-3\). Therefore, the y-intercept is \(-3\).

Step 3: Determining the Axis of Symmetry

The axis of symmetry for a quadratic function is given by the line \(x=h\). For this function, the axis of symmetry is \(x=-0.5\).

Step 4: Plotting the Graph and Calculating Domain and Range

First, plot the vertex, the y-intercept, and the x-intercepts on the graph. The shape of the function is a upward opening parabola as \(a=2>0\). The domain of a quadratic function is all real numbers so for this function domain is \(-\infty, \infty\). The range of a upward opening parabola is given by \([k, \infty)\), so for this function the range is \([-3.5, \infty)\].

Key Concepts

Vertex

In a quadratic function, the vertex is a crucial point that tells us where the graph changes direction. The vertex serves as either the minimum or maximum point of the parabola, depending on whether it opens upwards or downwards. For any quadratic function in the form of \(f(x) = ax^2 + bx + c\), you can find the vertex coordinates by using the formula:

• \( h = -\frac{b}{2a} \)
• \( k = f(h) \)

Here, \(h\) represents the x-coordinate, and \(k\) represents the y-coordinate of the vertex. This pair, \((h, k)\), gives the exact turning point of the parabola on the graph. In our example, the function \(f(x)=2x^2+4x-3\) has a vertex at \((-0.5, -3.5)\). This point tells us that the minimum point of the parabola sits at \(-0.5\) on the x-axis and \(-3.5\) on the y-axis.

Intercepts

Intercepts are where the graph of the quadratic function crosses the axes. Specifically, they include the x-intercepts and y-intercept.

• X-intercepts: These are the points where the graph crosses the x-axis, meaning the y-value is zero. To find them, set the quadratic equation \(f(x)=0\) and solve for \(x\). For our equation, solve \(2x^2 + 4x - 3 = 0\). The solutions are the x-intercepts.
• Y-intercept: This is the point where the graph crosses the y-axis, meaning the x-value is zero. Substitute \(x=0\) into the quadratic function to find \(f(0)\). In our function, \(f(0) = -3\), so the y-intercept is \((0, -3)\).

Intercepts are useful for graph sketching and can provide key points that guide the shape of the parabola.

Axis of Symmetry

The axis of symmetry is an important feature of a parabola. It is a vertical line that passes through the vertex, dividing the parabola into two mirror-image halves. For quadratic functions, this line has the equation \(x = h\).

• Why is it important? Knowing the axis of symmetry helps when graphing because it aids in quickly determining other points on the graph by reflection over this line.
• In our example of the function \(f(x) = 2x^2 + 4x - 3\), the axis of symmetry is found at \(x = -0.5\).

Thus, the line \(x = -0.5\) divides the parabola into two identical halves, highlighting the geometric balance found within quadratic functions.

Domain and Range

The domain and range are fundamental features that tell us the set of possible inputs (domain) and outputs (range) of the quadratic function.

• Domain: For quadratic functions, the domain is all real numbers: \((-\infty, \infty)\). This is because you can substitute any real number for \(x\) into the function and find a corresponding \(y\).
• Range: The range of a quadratic function depends on the vertex and the direction in which the parabola opens. If the parabola opens upwards (like in our example, where \(a = 2\)), the range starts at the y-coordinate of the vertex and goes to infinity. Thus, in our function, the range is \([-3.5, \infty)\), starting from the vertex's y-value, \(-3.5\), and extending indefinitely upwards.

Understanding the domain and range is vital for grasping the full vision of where the graph lies on the coordinate plane.