Problem 18 - Algebra 2 and Trigonometry

Question

a. Sketch the graph of \(y=x^{2}+6 x+2\) b. From the graph, estimate the roots of the function to the nearest tenth. c. Use the quadratic formula to find the exact values of the roots of the function. d. Express the roots of the function to the nearest tenth and compare these values to your estimate from the graph.

Step-by-Step Solution

Answer

Estimated roots from graph: \(-0.4\), \(-5.6\). Exact roots using formula: \(-0.4\), \(-5.6\). Graph estimates agree with calculations.

Step 1: Rewrite the Function

The function given is \( y = x^2 + 6x + 2 \). This is a quadratic function. We'll start by identifying the coefficients: \( a = 1 \), \( b = 6 \), and \( c = 2 \). These will be important for later steps.

Step 2: Sketch the Parabola

Since the coefficient of \( x^2 \) is positive, the parabola opens upwards. The vertex form of a quadratic is \( y = (x + \frac{b}{2})^2 - \frac{b^2}{4} + c \). Using \( b = 6 \), the vertex \( x \ ext{-coordinate} = -\frac{b}{2} = -3 \). Substitute to find \( y \) at this \( x \): \( y = (-3)^2 + 6(-3) + 2 = -7 \). Thus, the vertex is \( (-3, -7) \).

Step 3: Find Axis and Intercepts

The axis of symmetry is \( x = -3 \). The y-intercept, where \( x = 0 \), is \( y = 2 \). To find x-intercepts approximately, set \( y = 0 \) and solve \( x^2 + 6x + 2 = 0 \). We will refine this using the quadratic formula soon.

Step 4: Estimate Roots from the Graph

Plot the parabola with vertex \( (-3, -7) \) and y-intercept at \( (0, 2) \). By sketching, note where the curve crosses the x-axis. They look slightly less than \( -1 \) and slightly greater than \( -5 \). Estimate around \( x \approx -0.4 \) and \( x \approx -5.6 \) to the nearest tenth.

Step 5: Apply the Quadratic Formula

The quadratic formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Substitute values, \( a = 1 \), \( b = 6 \), and \( c = 2 \). Calculate \( b^2 - 4ac = 36 - 8 = 28 \). Thus, \( x = \frac{-6 \pm \sqrt{28}}{2} \). This simplifies to \( x = \frac{-6 \pm 2\sqrt{7}}{2} = -3 \pm \sqrt{7} \).

Step 6: Calculate Exact and Approximate Roots

Compute the roots approximately: \( -3 + \sqrt{7} \approx -0.354 \) and \( -3 - \sqrt{7} \approx -5.646 \). Rounded to the nearest tenth, they are \( x \approx -0.4 \) and \( x \approx -5.6 \).

Step 7: Compare with Graph Estimates

From the graph, the estimated roots were \( x \approx -0.4 \) and \( x \approx -5.6 \). From the quadratic formula, they are approximately \( x \approx -0.4 \) and \( x \approx -5.6 \), closely agreeing with the graphical estimate.

Key Concepts

Parabola

A parabola is a U-shaped curve that represents the graph of a quadratic function. When sketching a quadratic function like \( y = x^2 + 6x + 2 \), we first note the general form of a parabola, \( y = ax^2 + bx + c \). The shape and direction of the parabola are determined by the coefficient \( a \). For \( a > 0 \), the parabola opens upwards, and for \( a < 0 \), it opens downwards.
The given quadratic function has \( a = 1 \), so the parabola opens upwards. To get a good sketch, it's helpful to find key features such as the vertex and the axis of symmetry (which is a vertical line that passes through the vertex). In this case, after calculating, the vertex is at \((-3, -7)\). This gives a good starting point to draw the parabola, ensuring that it passes through this central point and creates the symmetrical U-shape.

Quadratic Formula

The quadratic formula is a powerful tool for finding the roots of any quadratic equation. It's expressed as \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here, \( a, b, \) and \( c \) are the coefficients from the quadratic equation \( ax^2 + bx + c = 0 \).
In the equation \( y = x^2 + 6x + 2 \), the coefficients are identified as \( a = 1 \), \( b = 6 \), and \( c = 2 \). Using these, you calculate the discriminant \( b^2 - 4ac \) which helps us to determine the nature of the roots. A positive discriminant indicates two distinct real roots, which is the case here with a value of 28.
After plugging these values into the formula, we get \( x = -3 \pm \sqrt{7} \), resulting in two specific roots, giving us exact solutions.

Roots of a Function

The roots of a function are the x-values where the function equals zero, essentially where the parabola crosses the x-axis. Identifying these root values involves solving the equation \( x^2 + 6x + 2 = 0 \). Using a graph, you can eyeball these points, but they can also be found accurately using the quadratic formula.
From the quadratic formula, the roots were calculated as \( -3 + \sqrt{7} \) and \( -3 - \sqrt{7} \), which approximate to \( -0.4 \) and \( -5.6 \) respectively when rounded to the nearest tenth. These roots tell us where the parabola intersects the x-axis, which are crucial in understanding the behavior of the graph.

Vertex of a Parabola

The vertex of a parabola is a key feature that marks its maximum or minimum point. For the parabola described by \( y = x^2 + 6x + 2 \), the vertex provides valuable symmetry and can be used to fully outline the parabola's shape.
To find the vertex, use the formula \( x = -\frac{b}{2a} \) to determine the x-coordinate. With \( b = 6 \) and \( a = 1 \), this yields \( x = -3 \). Substituting \( x = -3 \) back into the function, \( y = (-3)^2 + 6(-3) + 2 = -7 \), gives the vertex coordinate \((-3, -7)\).
Knowing the vertex helps in sketching the parabola accurately and provides the deepest point (minimum) of this specific U-shaped graph. It is important as it dictates the overall shape and direction of the parabola and is a foundation for finding other attributes, like the axis of symmetry.