Problem 13 - Algebra 1: Concepts and Skills

Question

Sketch the graph of the function. Label the coordinates of the vertex. Write an equation for the axis of symmetry. $$ y=-3 x^{2}+6 x+2 $$

Step-by-Step Solution

Answer

The vertex is at the point (-1,-7) and the equation of the axis of symmetry is x = -1.

Step 1: Find the vertex

Use the formula for finding the vertex of a parabola, which is (-b/2a, f(-b/2a)). Here, b=6, a=-3. So, the x-coordinate of the vertex is -(-6/2*-3) = -1. Substitute x=-1 into the function -3x^2 + 6x + 2 to get the y-coordinate of the vertex, which will be -3*(-1)^2 + 6*-1 + 2 = -3 - 6 + 2 = -7.

Step 2: Find the equation of the axis of symmetry

The axis of symmetry of a parabola given by y=ax^2 + bx + c is x=-b/2a. Here a=-3 and b=6. So, the equation of the axis of symmetry is x=-(-6/2*-3) = -1.

Step 3: Sketch the graph

Plot the vertex point (-1, -7) from step 1 on the graph. Also plot the axis of symmetry from step 2 x = -1. Know that a parabola opens upwards if a > 0 and opens downwards if a < 0. Here a=-3, so the parabola will open downwards. Sketch the graph using these details.

Key Concepts

Vertex of a parabola

The vertex of a parabola is a critical point. It acts like the "tip" or "turning point" of the parabola. Finding the vertex allows us to understand the position and the minimum or maximum point of the graph, depending on whether it opens up or down.
To find the vertex, use the formula \[(-\frac{b}{2a}, f(-\frac{b}{2a}))\]where "a" and "b" are coefficients from the quadratic equation in the form \[y = ax^2 + bx + c\].
In the example given with the equation \[y = -3x^2 + 6x + 2\]we have \[b = 6\] and \[a = -3\].
- Calculate the x-coordinate: \(-\frac{b}{2a} = -\frac{6}{2 imes (-3)} = -1\)
- To find the y-coordinate, substitute the x-value back into the equation: \[-3(-1)^2 + 6(-1) + 2 = -7\].

• Thus, the vertex coordinates are \((-1, -7)\).
• The vertex represents the maximum point here since "a" is negative, and the parabola opens downwards.

Understanding how to find the vertex can greatly simplify graphing and analyzing quadratic functions.

Axis of symmetry

The axis of symmetry of a parabola is an imaginary vertical line that divides the parabola into two mirror-image halves. It passes through the vertex and is crucial for understanding the symmetry in the quadratic function.
To find the axis of symmetry:

• Use the formula \(x = -\frac{b}{2a}\)
• Similar to determining the x-coordinate of the vertex, this formula provides the x-value where the symmetry occurs.

In the equation \[y = -3x^2 + 6x + 2\], the axis of symmetry calculation will involve:
- \(-\frac{6}{2 imes (-3)} = -1\)
Therefore, the axis of symmetry for the given quadratic is \(x = -1\).
This line helps in sketching the parabola as it shows where it is symmetric around. When graphing, each point on one side of this line has a mirror point on the opposite side.

Parabola graphing

Graphing a parabola involves using key features like the vertex and the axis of symmetry to create an accurate representation on the coordinate plane. Follow these simple steps to make graphing less daunting.
First, plot the vertex point found previously. As in this case with \((-1, -7)\), this point is plotted first.
Next, sketch the axis of symmetry. It's a vertical line at \(x = -1\) in this instance. Use it as a guide to help maintain balance in the graph.
Then, consider the "a" coefficient from \[y = -3x^2 + 6x + 2\]. A negative "a" indicates that the parabola opens downward. If it were positive, the parabola would open upwards.

• Graph additional points by choosing x-values on either side of the axis of symmetry and calculating corresponding y-values through substitution.
• Mirror these points across the axis of symmetry for added accuracy.

Finally, smoothly connect these points to complete the parabola. Each characteristic of the quadratic we've used will ensure the graph accurately reflects the function's behavior.