Problem 33 - Algebra for College Students

Question

Find the vertex for the graph of each quadratic function. $$f(x)=-2 x^{2}+20 x+1$$

Step-by-Step Solution

Answer

The vertex is (5, 51).

Step 1: Identify the coefficients

The general form of a quadratic function is given by \( f(x) = ax^2 + bx + c \). Here, the coefficients are \( a = -2 \), \( b = 20 \), and \( c = 1 \).

Step 2: Calculate the x-coordinate of the vertex

The x-coordinate of the vertex can be found using the formula \( x = -\frac{b}{2a} \). Substituting the values of \( b \) and \( a \), we have: \[ x = -\frac{20}{2(-2)} = -\frac{20}{-4} = 5 \]

Step 3: Calculate the y-coordinate of the vertex

Substitute \( x = 5 \) back into the quadratic function to find the y-coordinate: \[ f(5) = -2(5)^2 + 20(5) + 1 \] Calculate the square and multiplication: \[ f(5) = -2(25) + 100 + 1 = -50 + 100 + 1 = 51 \] So the y-coordinate is \( 51 \).

Step 4: State the vertex

The vertex of the quadratic function \( f(x) = -2x^2 + 20x + 1 \) is \( (5, 51) \).

Key Concepts

quatdratic function

A quadratic function is a type of polynomial that is typically written in the form \[ f(x) = ax^2 + bx + c, \] where \( a \), \( b \), and \( c \) are constants. Quadratic functions graph as parabolas, which are U-shaped curves, and the direction of the parabola's opening depends on the value of \( a \).

• If \( a > 0 \), the parabola opens upwards.
• If \( a < 0 \), the parabola opens downwards.

The highest or lowest point of the parabola is known as the vertex. Understanding the structure of a quadratic function is essential for analyzing its graph and properties.

vertex calculation

The vertex of a quadratic function is a crucial feature, as it represents the highest or lowest point on its graph. To find the vertex, follow these steps: First, identify the coefficients \(a\), \(b\), and \(c\) from the standard form \(f(x) = ax^2 + bx + c\). For example, in the function \(f(x) = -2x^2 + 20x + 1\), the coefficients are \(a = -2\), \(b = 20\), and \(c = 1\). Next, calculate the x-coordinate of the vertex using the formula: \[x = -\frac{b}{2a}. \] Substituting in the values from our example, we get: \[x = -\frac{20}{2(-2)} = \frac{20}{4} = 5. \] Finally, find the y-coordinate by substituting the x-coordinate back into the quadratic function: \(f(5) = -2(5)^2 + 20(5) + 1 = -50 + 100 + 1 = 51. \) So, the vertex of the function \(f(x) = -2x^2 + 20x + 1\) is \((5, 51)\).

parabola vertex

The vertex of a parabola is a key point that represents either its maximum or minimum value. For the quadratic function, it is the point \((x, y)\) at which the function reaches its highest or lowest value depending on the direction of the parabola's opening. - If the parabola opens upwards (\(a > 0\)), the vertex represents the minimum value. - If the parabola opens downwards (\(a < 0\)), the vertex represents the maximum value. The vertex is also the point of symmetry for the parabola. This means the parabola is mirror-symmetrical about the vertical line that passes through the vertex, often referred to as the axis of symmetry. Again, the x-coordinate of the vertex is given by \(x = -\frac{b}{2a}\), ensuring you have the correct position before finding its corresponding y-value. The vertex helps in sketching the graph and understanding the function's range and domain.