Problem 3 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

For Problems 1 through 8, graph the function. Label the \(x\) - and \(y\) -intercepts and the coordinates of the vertex. $$ f(x)=x^{2}+5 x-6 $$

Step-by-Step Solution

Answer

The function \(f(x)=x^{2}+5x-6\) graph has \(x\)-intercepts at \(x = 1\) and \(x = -6\), a \(y\)-intercept at \(y = -6\), and the vertex at \((-2.5, -6.25)\).

Step 1: Determine the \(x\)-intercepts

By setting \(f(x) = 0\), the equation becomes \(x^2 + 5x - 6 = 0\). By factorising this we get \((x - 1)(x + 6) = 0\). Thus, the \(x\)-intercepts are \(x = 1\) and \(x = -6\).

Step 2: Determine the \(y\)-intercept

By setting \(x = 0\) in \(f(x)\), we find that \(f(0) = -6\). Thus, the \(y\)-intercept is \(y = -6\).

Step 3: Determine the vertex

The formula to find the \(x\)-coordinate of the vertex is \(x = -b / 2a\). In our equation, \(a = 1\) and \(b = 5\). Substituting these values, we get \(x = -5 / 2\) or \(x = -2.5\). Plugging this back into the equation \(f(x) = x^2 + 5x - 6\) to find the corresponding \(y\)-coordinate of the vertex gives us \(y = (-2.5)^2 + 5*(-2.5) - 6= -6.25 \). Thus, the vertex of the function is \((-2.5, -6.25)\).

Step 4: Plot the function

Now, with the \(x\)-intercepts at \(x = 1\) and \(x = -6\), \(y\)-intercept at \(y = -6\) and the vertex at \((-2.5, -6.25)\), a plot of the function can be made. It's important to keep in mind that since the coefficient of \(x^2\) is positive, the parabola opens upwards.

Key Concepts

Graphing Parabolas

Parabolas are a type of curve shaped like a "U" or an inverted "U". They are the graphical representation of quadratic functions of the form \(f(x) = ax^2 + bx + c\). In graphing a parabola, identifying key points such as intercepts and the vertex is crucial for accurate plotting.

• The direction the parabola opens depends on the sign of the coefficient \(a\):
• If \(a > 0\), the parabola opens upwards, resembling a smile.
• If \(a < 0\), it opens downwards, like a frown.

To graph a parabola accurately:- Determine the vertex, which is the highest or lowest point on the graph.- Find the \(x\)- and \(y\)-intercepts.- Plot the vertex and the intercepts on the coordinate plane.- Draw a smooth curve through these points, keeping in mind the direction indicated by \(a\).
The symmetry of the parabola is about its vertical line through the vertex, known as the axis of symmetry.

Vertex of a Parabola

The vertex of a parabola is a critical point that indicates the maximum or minimum value of the function, depending on whether the parabola opens downwards or upwards, respectively. It is the point where the parabola changes direction.
To find the vertex of a parabola given in the standard quadratic form \(f(x) = ax^2 + bx + c\), you can use the formula:\[x = -\frac{b}{2a}\]
This formula helps you find the \(x\)-coordinate of the vertex. Once you have this value, substitute it back into the original equation to find the \(y\)-coordinate. For the function \(f(x) = x^2 + 5x - 6\), the \(x\)-coordinate of the vertex is:\[x = -\frac{5}{2 \times 1} = -2.5\]
And by finding \(f(-2.5)\), you calculate the \(y\) as:\[f(-2.5) = (-2.5)^2 + 5(-2.5) - 6 = -6.25\]
Thus, the vertex is \((-2.5, -6.25)\). The vertex is a powerful tool as it helps in sketching the parabola and in understanding the function's behavior.

Intercepts of a Quadratic Function

Intercepts are points where the graph intersects the axes. In quadratic functions, identifying intercepts helps in understanding and sketching the parabola.

X-Intercepts

The \(x\)-intercepts are the points where the parabola cuts the \(x\)-axis. Set \(f(x) = 0\) to find these points. Solving the equation \(x^2 + 5x - 6 = 0\) by factorizing gives:\[(x - 1)(x + 6) = 0\]
This results in \(x = 1\) and \(x = -6\). Thus, the \(x\)-intercepts are at these coordinates.

Y-Intercept

The \(y\)-intercept is where the parabola crosses the \(y\)-axis. Set \(x = 0\) in the equation to find it. For \(f(x) = x^2 + 5x - 6\), substituting yields:\[f(0) = 0^2 + 5 \, \times \, 0 - 6 = -6\]
Thus, the \(y\)-intercept is \(y = -6\).Knowing these points aids in plotting the parabola accurately and highlights the roots and initial value of the function.