Problem 44 - Algebra 1: Concepts and Skills

Question

Sketch the graph of the function. Label the coordinates of the vertex. $$ y=-3 x^{2}-3 x+4 $$

Step-by-Step Solution

Answer

The graph of the function \(y = -3x^2 -3x + 4\) is a downward opening parabola with vertex at (0.5, 2.5).

Step 1: Find the Vertex

The vertex form of a quadratic function is \(y = a(x-h)^2 + k\), where (h, k) are the vertex coordinates. To find 'h', use the formula \(h = - \frac{b}{2a}\). Here, a = -3 and b = -3. So, \(h = - \frac{-3}{2*(-3)} = 0.5\). Substitute 'h' into the function to find 'k': \(k = -3*(0.5)^2-3*0.5+4 = 3 - 1.5 + 4 = 2.5\). Thus, vertex = (0.5, 2.5).

Step 2: Find the Axis of Symmetry

The axis of symmetry is the vertical line \(x = h\), which here is \(x = 0.5\).

Step 3: Sketch the Graph

Start with the vertex (0.5, 2.5) and the symmetry axis \(x = 0.5\). Because 'a' is negative, the parabola opens downwards. Now, draw the parabola symmetrically on both sides of the symmetry axis.

Key Concepts

Vertex of a Parabola

In a quadratic function, the vertex plays a pivotal role as it represents the maximum or minimum point of the parabola. This is where the function reaches an extremum. To find the vertex of the parabola given by the equation \( y = -3x^2 - 3x + 4 \), we use the vertex formula:

• The standard form of a quadratic is \( y = ax^2 + bx + c \).
• The vertex form is \( y = a(x-h)^2 + k \).
• Here, the vertex \((h, k)\) can be found using \( h = -\frac{b}{2a} \).

Plugging in the values \( a = -3 \) and \( b = -3 \), we find \( h = 0.5 \). To find \( k \), substitute \( h \) back into the equation to get \( k = -3(0.5)^2 - 3(0.5) + 4 = 2.5 \). Thus, the vertex is \( (0.5, 2.5) \), acting as the highest point since the parabola opens downwards due to the negative \( a \) value.

Axis of Symmetry

The axis of symmetry is a vital feature in understanding the symmetry of a parabola. It is an imaginary line that vertically splits the parabola into two mirror-image halves. For any quadratic function, the axis of symmetry can be easily determined from the vertex. It is always given by the equation \( x = h \).So, for the function \( y = -3x^2 - 3x + 4 \), after determining the vertex \((0.5, 2.5)\), we identify its axis of symmetry as the line \( x = 0.5 \). This line helps when sketching the graph, ensuring that all points on the parabola are at equal horizontal distances from this axis. This property makes grouping and calculating points for the parabola more systematic and organized.

Graphing Parabolas

Graphing parabolas involves several steps that bring together various components of the quadratic function. It helps visually represent different aspects, like how it opens and its direction. Here's a simple way to approach it:

• Start by plotting the vertex, which is the point \( (0.5, 2.5) \) in this scenario.
• Draw the axis of symmetry at \( x = 0.5 \), ensuring the parabola is evenly balanced around this vertical line.
• The parabola opens downward since the coefficient \( a = -3 \) is negative. This means its arms point towards the negative \( y \)-axis.
• Finally, choose additional points on either side of the vertex to sketch a smooth, curved line symmetric around the axis to complete the graph.

Remember, the more accurately you can plot the points around the vertex and axis of symmetry, the more precise your graph of the parabola will be.