Problem 28 - 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. $$ f(x)=x^{2}-2 x-15 $$

Step-by-Step Solution

Answer

Axis of symmetry is given by \(x=1\). The domain of the function \(f(x)\) is all real numbers, and the range is \(y \leq -16\).

Step 1: Find the Vertex and Intercepts

To find the vertex of this quadratic equation, we use the formula \( -b / 2a \), where \(a\) and \(b\) are coefficients in the equation \( f(x)=ax^{2}+bx+c \). In this case, \(a=1\) and \(b=-2\), so the vertex is \( x=-(-2)/2*1=1 \). We substitute \(x=1\) to find the \(y\) value for the vertex, hence \(y=(1)^{2}-2*(1)-15=-16\). Hence the vertex is (1, -16). The intercepts can be found by setting \( f(x)=0 \), and solve for \( x \). This gives \( 0 = x^{2} - 2x - 15 = (x-5)(x+3) \), hence \(x_1 = 5\) and \(x_2 = -3\) are the x-intercepts.

Step 2: Plot the Vertex and Intercepts and Sketch the Parabola

Plot the vertex (1, -16) and the intercepts (5, 0) and (-3, 0) on a graph. The x-axis of symmetry will pass through the vertex, and the curve of the parabola will be symmetrical with respect to this axis. So sketch the curve by taking advantage of this symmetry.

Step 3: Determine Axis of Symmetry, Domain and Range

The equation of the axis of symmetry is \( x=1 \), which we derived from the x-value of the vertex. The domain is all real numbers because the parabola extends indefinitely to the left and right. The range is all values less than or equal to the y-value of the vertex, which is -16. Hence the range is \(y \leq -16\).

Key Concepts

Vertex of a Parabola

Understanding the vertex is crucial in quadratic functions. The vertex of a parabola is a specific point where the curve changes direction. It can be seen as the 'turning point' and is either the highest or lowest point on the graph. To find the vertex, we use the formula \(-\frac{b}{2a}\), derived from the standard form of the quadratic equation \(f(x) = ax^2 + bx + c\). Here, \(a\) and \(b\) refer to the coefficients of the equation.

• For the equation \(f(x) = x^2 - 2x - 15\), substituting \(a = 1\) and \(b = -2\) into the formula gives \(x = 1\).

We then substitute \(x = 1\) back into the equation to find the \(y\)-value, resulting in \(y = -16\).Therefore, the vertex is the point \((1, -16)\), marking the lowest point of the parabola in this instance.

Intercepts of a Quadratic

The intercepts of a quadratic are the points where the parabola crosses the axes.

• The x-intercepts occur when the output \(f(x) = 0\).
• To find these, we set the quadratic equal to zero and solve for \(x\).

For our equation \(x^2 - 2x - 15\), we factor it as \((x-5)(x+3)\). Solving for \(x\), we find the intercepts at \(x = 5\) and \(x = -3\).The y-intercept is found by evaluating \(f(x)\) at \(x = 0\). For this function, substituting \(x = 0\) gives \(f(0) = -15\). Therefore, the y-intercept is at \((0, -15)\). These intercepts provide critical points that help sketch the graph of the function.

Axis of Symmetry

The axis of symmetry is a line that divides the parabola into two mirror-image halves. It passes through the vertex and helps to ensure the graph is accurately represented.In our case, since the vertex is \((1, -16)\), the axis of symmetry is the vertical line \(x = 1\). Each point on the parabola has a mirror point on the opposite side of this line. This symmetrical property is helpful when sketching the parabola. The equation \(x = -\frac{b}{2a}\) not only helps in finding the vertex but also gives the equation for the axis of symmetry directly.

Domain of a Function

The domain of a function refers to all possible input values (\(x\)-values) that the function can accept. For quadratic functions, the domain is generally all real numbers, \(-\infty < x < +\infty\).This is because the parabola extends indefinitely in both the positive and negative directions along the x-axis.

• Hence, for our function \(f(x) = x^2 - 2x - 15\), the domain is all real numbers.

This broad domain allows for a wide range of input values, which is characteristic of all quadratic functions.

Range of a Function

The range of a function is the set of possible output values (\(y\)-values) the function can produce. For quadratics, this depends on whether they open upwards or downwards.Since our parabola opens upwards, the lowest point is the vertex. This means that all \(y\)-values are equal to or greater than the vertex's \(y\)-value. For the function \(f(x) = x^2 - 2x - 15\), the vertex is at \((1, -16)\). Thus, the range includes all real numbers \(y\leq-16\). Understanding the range helps in visualizing the extent of the parabola along the y-axis.