Problem 16 - Beginning and Intermediate Algebra

Question

Find the vertex of each parabola. For each equation, decide whether the graph opens up, down, to the left, or to the right, and whether it is wider, narrower, or the same shape as the graph of \(y=x^{2}\). If it is a parabola with a vertical axis of symmetry, find the discriminant and use it to determine the number of \(x\) -intercepts. $$ f(x)=-x^{2}+7 x+2 $$

Step-by-Step Solution

Answer

The vertex is (3.5, 14.25), the parabola opens downward, it has the same shape as \( y = x^2 \), and there are two distinct \( x \) intercepts due to a positive discriminant.

Step 1: Identify the Equation Form

The given equation is in the form of a quadratic function: \[ f(x) = ax^2 + bx + c \] Here, \( a = -1 \), \( b = 7 \), and \( c = 2 \).

Step 2: Determine the Vertex

The vertex ((x_h, y_k)) of a parabola \( y = ax^2 + bx + c \) can be found using the formula:\( x_h = -\frac{b}{2a} \).Substituting the values for \( a\) and \( b \): \( x_h = -\frac{7}{2(-1)} = \frac{7}{2} = 3.5 \).Compute \( y_k\) by evaluating \( f(3.5) \): \( f(3.5) = - (3.5)^2 + 7 (3.5) + 2 = -12.25 + 24.5 + 2 = 14.25 \).Therefore, the vertex is \( (3.5, 14.25) \).

Step 3: Determine the Direction of the Graph

Since \(a = -1\) (negative), the parabola opens downward.

Step 4: Compare the Shape to \( y = x^2 \)

The coefficient \( |a| \) is 1, which is the same as \( y = x^2 \). Thus, the parabola is the same shape as \( y = x^2 \).

Step 5: Find the Discriminant

The discriminant \( \Delta \) of a quadratic equation \( ax^2 + bx + c \) is given by:\[ \Delta = b^2 - 4ac \].Substitute the values of \( a, b, c \): \( \Delta = 7^2 - 4(-1)(2) = 49 + 8 = 57 \).Since the discriminant is positive (>0), there are two real and distinct \( x \) intercepts.

Key Concepts

quadratic function

A quadratic function is a type of polynomial function that can be represented by the equation: \[ f(x) = ax^2 + bx + c \]. Here, \(a\), \(b\), and \(c\) are constants, and \(x\) represents the variable. Quadratic functions are notable for their parabolic graphs that have a U-shape, which can either open upwards or downwards.
The key features of a quadratic function include:

• The coefficient \(a\) determines the direction the parabola opens.
• The term \(bx\) affects the position of the vertex and axis of symmetry.
• The term \(c\) represents the y-intercept, where the graph crosses the y-axis.

Quadratic functions are common in various real-world applications like physics, engineering, and economics.

vertex of a parabola

The vertex of a parabola is the point where the curve changes direction, representing the maximum or minimum value of the function, depending on whether it opens upwards or downwards. For a quadratic function in the form \( y = ax^2 + bx + c \), the vertex can be found using the formula: \[ x_h = -\frac{b}{2a} \].
Substituting this \(x_h\) value back into the function helps us find \(y_k\), the corresponding y-coordinate of the vertex.
In our exercise, with \( a = -1 \) and \( b = 7 \): \( x_h = -\frac{7}{2(-1)} = 3.5 \).
Evaluating \( f(3.5) \) gives: \[ f(3.5) = - (3.5)^2 + 7(3.5) + 2 = -12.25 + 24.5 + 2 = 14.25 \].
Thus, the vertex of the parabola \( f(x) = -x^2 + 7x + 2 \) is \( (3.5, 14.25) \). The vertex is crucial for understanding the graph's shape and position.

discriminant

The discriminant of a quadratic equation \( ax^2 + bx + c \) helps determine the nature of its roots and is given by the formula: \[ \Delta = b^2 - 4ac \]. The value of the discriminant reveals the following:

• If \( \Delta > 0 \), there are two distinct real roots.
• If \( \Delta = 0 \), there is one real root (a repeated or double root).
• If \( \Delta < 0 \), there are no real roots, only complex roots.

In our exercise, substituting \( a = -1 \), \( b = 7 \), and \( c = 2 \) into the formula gives: \[ \Delta = 7^2 - 4(-1)(2) = 49 + 8 = 57 \]. Since \( \Delta > 0 \), this indicates that the quadratic function \( f(x) = -x^2 + 7x + 2 \) has two distinct real roots. Knowing the discriminant also helps predict the number of x-intercepts, offering insights into the graph's interaction with the x-axis.