Problem 58 - Algebra 1

Question

SKETCHING GRAPHS Sketch the graph of the function. Label the vertex. $$ y=-4 x^{2}+32 x-20 $$

Step-by-Step Solution

Answer

The vertex of the function \(y=-4x^2+32x-20\) is at the point (4, 4). The parabola opens downward.

Step 1: Identify the coefficients

The coefficients from the function are: \(a = -4\), \(b = 32\), and \(c = -20\).

Step 2: Calculate the vertex

Use the formula to find the x-coordinate of the vertex \(x = -b/2a\), so \(x = -32/(2*(-4)) = 4\). To find the y-coordinate of the vertex, substitute the x-value into the function: \(y = -4*4^2 + 32*4 - 20 = 4\). Thus, the coordinates of the vertex are (4, 4).

Step 3: Plot the vertex and draw the graph

Plot the vertex at the point (4, 4). As the coefficient of \(x^2\) is negative, the parabola opens downward. At \(x = 0\), \(y = -20\). And at \(x = 8\), the function also equals \(y = -20\). So plot these points and then sketch the graph by connecting the points. The graph is symmetric with respect to the line \(x = 4\).

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a crucial point, representing the peak or the lowest point of the parabola, depending on its orientation. In the equation \[y = ax^2 + bx + c\]the vertex can be found using the formula for the x-coordinate:\[x = \frac{-b}{2a}\]Substituting this x-value back into the original equation yields the y-coordinate.

• For example, with the function \(y = -4x^2 + 32x - 20\), the coefficients are \(a = -4\), \(b = 32\), and \(c = -20\). Using the formula, we calculate:\[x = \frac{-32}{2(-4)} = 4\]
• Substituting \(x = 4\) back into the equation gives us the y-value:\[y = -4(4)^2 + 32(4) - 20 = 4\]
• So, the vertex of this parabola is at the point \((4, 4)\).

This vertex tells us that the parabola reaches its highest point at \((4, 4)\) since the parabola opens downwards.

Graphing Parabolas

Graphing a parabola involves plotting points on a coordinate plane and connecting them in the shape of a curve. The key points include the vertex and other crucial points like the y-intercept and additional points to define the shape.
The process includes:

• Plotting the Vertex: Start by plotting the vertex, which in this case is \((4, 4)\).
• Finding Intercepts: The y-intercept is where \(x = 0\). Using the equation with \(x = 0\), we find:\[y = -4(0)^2 + 32(0) - 20 = -20\]
• Symmetry Points: Since the parabola is symmetric about its vertex, find additional points for clarity. At \(x = 8\), the y-value is equal to the y-intercept's y-value \(-20\), so we plot \((8, -20)\).
• Drawing the Curve: Connect these key points smoothly with a curve that opens downward.

With the vertex and symmetry points plotted, sketch the parabola to reflect its accurate shape.

Axis of Symmetry

The axis of symmetry of a parabola is a vertical line that divides the parabola into two mirror-image halves. It passes through the vertex and plays a vital role in understanding the graph’s symmetry.
The equation for the axis of symmetry is:\[x = \frac{-b}{2a}\]

• For the function \(y = -4x^2 + 32x - 20\), the axis of symmetry is calculated to be \(x = 4\).
• This means the line \(x = 4\) is where you could fold the parabola so each half would match the other.
• This line helps in quickly finding points equidistant from the vertex, simplifying graphing.

Understanding this concept helps not only in graphing but also in predicting the behavior of the quadratic function.