Problem 30 - College Algebra

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. \(f(x)=2 x^{2}-7 x-4\)

Step-by-Step Solution

Answer

The vertex is at (1.75, -5.125). The x-intercepts are at x = 4 and x = -0.5, and the y-intercept is at (0, -4). The axis of symmetry is at x = 1.75. The domain is \( (-\infty, +\infty)\). The range is \([-5.125, +\infty)\).

Step 1: Finding Vertex

The vertex of a quadratic function is given by \((-b/2a, f(-b/2a))\). Plug \(a=2\) and \(b=-7\) into the formula to find the x-coordinate of the vertex and then substitute this into the equation to find the y-coordinate: \(x=-(-7)/(2*2)=7/4=1.75\). \(f(1.75)=2(1.75)^{2}-7(1.75)-4=-5.125\). Therefore, the vertex is (1.75, -5.125).

Step 2: Finding Intercepts

The x-intercepts are the roots of the equation, meaning where \(f(x) = 0\). Setting \(f(x) = 0\), we have \(2x^{2}-7x-4=0\). Solving this quadratic equation, we use the formula \(x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\). Thus, \(x= \frac{-(-7) \pm \sqrt{(-7)^{2}-4*2*(-4)}}{2*2}\). After calculation, we get \(x=4, -0.5\) which are the x-intercepts. For y-intercept, set x=0; thus, \(f(0) = 2*0^{2}-7*0-4 = -4\) so the y intercept is (0,-4).

Step 3: Finding Axis of Symmetry

The Axis of Symmetry of a parabola is a vertical line that passes through the vertex. It has the equation \(x = -b/2a\), thus in this case, the axis of symmetry is \(x=1.75\).

Step 4: Determining Domain and Range

Domain refers to the set of all possible x-values. For any quadratic function, the domain is all real numbers. Therefore, the domain is \( (-\infty, +\infty)\). The range represents all possible output (y-values) of the function. As this is a upward opening parabola (a > 0), the minimum y-value is the y-coordinate of the vertex. Therefore, the range is \([-5.125, +\infty)\).

Key Concepts

Vertex

The vertex of a quadratic function is like the tip of the parabola. It is a significant point that shows the highest or lowest point, depending on whether the parabola opens upwards or downwards. For the quadratic function given by the equation \(f(x) = 2x^2 - 7x - 4\), you can find the vertex using the formula:

• The x-coordinate is calculated by \(-\frac{b}{2a}\).
• The y-coordinate is found by plugging this x value back into the function \(f\).

In this case, plugging in the values of \(a = 2\) and \(b = -7\) gives us an x-coordinate of 1.75. Substituting \(1.75\) back into the function gives us a y-coordinate of \(-5.125\). Hence, the vertex is at the point \((1.75, -5.125)\). This vertex indicates that the parabola reaches its lowest value at this point.

Intercepts

Intercepts are the points where the graph crosses the axes. They tell us about the roots and the initial value of the function. - **X-intercepts**: These are where the graph crosses the x-axis, meaning the function value \(f(x) = 0\). For the given quadratic, solve the equation \(2x^2 - 7x - 4 = 0\) for \(x\). Use the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] Substituting \(a = 2\), \(b = -7\), and \(c = -4\), we find the x-intercepts to be \(x = 4\) and \(x = -0.5\). - **Y-intercept**: This is where the graph crosses the y-axis, found by setting \(x = 0\). For our function, \(f(0) = -4\). Thus, the y-intercept is at \((0, -4)\). These intercepts help draw a more accurate graph of the function.

Axis of Symmetry

In a quadratic function, the axis of symmetry acts like a mirror line that runs through the vertex and divides the parabola into two identical halves. The formula to find this line is exactly the same as the formula for finding the x-coordinate of the vertex, which is given by \(x = -\frac{b}{2a}\).For our specific function \(f(x) = 2x^2 - 7x - 4\) with \(a = 2\) and \(b = -7\), the axis of symmetry is the vertical line \(x = 1.75\). This line marks the position of the vertex and helps guide you in sketching out both sides of the parabola symmetrically on the graph. Essentially, it tells you exactly where to fold the parabola in half.

Domain and Range

Every graph has boundaries known as domain and range. These boundaries tell us how far the graph stretches on both the x-axis and y-axis. - **Domain**: The domain of a quadratic function is always all real numbers, \( (-\infty, \infty) \). This means the parabola extends infinitely in both the left and right directions along the x-axis, and there are no restrictions on the x-values the function can take.- **Range**: The range, however, depends on whether the parabola opens upwards or downwards. For our function \(f(x) = 2x^2 - 7x - 4\), the parabola opens upwards (since \(a = 2 > 0\)). Therefore, the lowest point of the parabola is the y-value at the vertex. Here, the vertex happens at \(y = -5.125\). Thus, the range of the function is \([-5.125, \infty)\), encompassing all y-values greater than or equal to \(-5.125\). This tells us that the parabola never dips below \(-5.125\) on the y-axis.