Problem 19 - Algebra and Trigonometry

Question

A quadratic function is given. (a) Express the quadratic function in standard form. (b) Find its vertex and its x- and y-intercept(s). (c) Sketch its graph. $$ f(x)=2 x^{2}-20 x+57 $$

Step-by-Step Solution

Answer

The vertex is (5, 7), no real x-intercepts, y-intercept is (0, 57).

Step 1: Express in Standard Form

The quadratic function is already in standard form, which is written as \( ax^2 + bx + c \). In this case, \( a = 2 \), \( b = -20 \), and \( c = 57 \). Therefore, \( f(x) = 2x^2 - 20x + 57 \) is already in the required form.

Step 2: Find the Vertex

To find the vertex of a parabola given by \( ax^2 + bx + c \), use the vertex formula \( x = \frac{-b}{2a} \). Here, \( a = 2 \) and \( b = -20 \). So, \( x = \frac{-(-20)}{2 \cdot 2} = \frac{20}{4} = 5 \). Substitute \( x = 5 \) back into the function to find \( y \):\[ f(5) = 2(5)^2 - 20(5) + 57 = 50 - 100 + 57 = 7 \]. Thus, the vertex is \((5, 7)\).

Step 3: Find the x-intercepts

To find the x-intercepts, set \( f(x) = 0 \) and solve for \( x \):\[ 2x^2 - 20x + 57 = 0 \]. Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 2 \), \( b = -20 \), and \( c = 57 \). Calculate the discriminant: \( b^2 - 4ac = (-20)^2 - 4 \cdot 2 \cdot 57 = 400 - 456 = -56 \). Since the discriminant is negative, there are no real x-intercepts.

Step 4: Find the y-intercept

The y-intercept is found by setting \( x = 0 \) in the quadratic function. So, \( f(0) = 57 \). Thus, the y-intercept is \((0, 57)\).

Step 5: Sketch the Graph

Now, combine all the information gathered: the vertex at \((5, 7)\), no x-intercepts, and the y-intercept at \((0, 57)\). Since \( a = 2 > 0 \), the parabola opens upwards. Plot the vertex and the y-intercept, and sketch a smooth curve that makes a U-shape upwards.

Key Concepts

Vertex

In a quadratic function, the vertex represents the highest or lowest point of the parabola. It's a key feature that helps in sketching the curve of the parabola.
The formula to find the x-coordinate of the vertex is \( x = \frac{-b}{2a} \). For the given function, \( f(x) = 2x^2 - 20x + 57 \), the values are \( a = 2 \) and \( b = -20 \).
Substituting these into the formula gives \( x = \frac{20}{4} = 5 \).
To find the y-coordinate, substitute \( x = 5 \) back into the function: \( f(5) = 2 \times 5^2 - 20 \times 5 + 57 = 7 \).
Therefore, the vertex is \((5, 7)\), indicating the minimum point of this upward-opening parabola.

X-intercept

The x-intercepts of a quadratic function are the points where the graph crosses the x-axis.
These can be found by setting \( f(x) = 0 \) and solving the quadratic equation. For \( f(x) = 2x^2 - 20x + 57 \), use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a = 2 \), \( b = -20 \), and \( c = 57 \).
Calculating the discriminant, \( b^2 - 4ac = 400 - 456 = -56 \), shows the value under the square root is negative.
Since the discriminant is negative, there are no real x-intercepts. Hence, the parabola does not cross the x-axis.

Y-intercept

The y-intercept of a quadratic function is where the graph crosses the y-axis.
This occurs when \( x = 0 \) in the equation \( f(x) = ax^2 + bx + c \).
For the given function, \( f(0) = 2 \times 0^2 - 20 \times 0 + 57 = 57 \).
Therefore, the y-intercept is at the point \((0, 57)\).
This is a key feature when graphing the function, as it shows where the parabola crosses the y-axis.

Parabola

A parabola is a symmetrical, curved shape that is the graph of a quadratic function.
For the quadratic function \( f(x) = 2x^2 - 20x + 57 \), the parabola opens upwards because \( a = 2 \) is positive.
Important features of this parabola include:

• The vertex at \((5, 7)\)
• No x-intercepts, as confirmed by the negative discriminant
• The y-intercept at \((0, 57)\)

These characteristics help in sketching the parabola, which is U-shaped and symmetrical around the vertex.

Standard Form

The standard form of a quadratic function is expressed as \( ax^2 + bx + c \). It helps identify crucial properties of the function swiftly.
For the equation \( f(x) = 2x^2 - 20x + 57 \), it is already in standard form.
From here, we identify:

• \( a = 2 \) - determines the parabola's direction (upward in this case)
• \( b = -20 \) - influences the slope of the parabola
• \( c = 57 \) - indicates the y-intercept

This form allows for straightforward calculations of the vertex, intercepts, and sketching of the graph.