Problem 15 - 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. $$ f(x)=-x^{2}+5 x+3 $$

Step-by-Step Solution

Answer

The vertex is \( \left( \frac{5}{2}, \frac{37}{4} \right) \). The graph opens downwards, has the same width as \( y = x^2 \), and has 2 \( x \)-intercepts.

Step 1: Identify the coefficients

Identify the coefficients in the quadratic equation. The standard form is given by \[ f(x) = ax^2 + bx + c \] Here, \( a = -1 \), \( b = 5 \), and \( c = 3 \).

Step 2: Determine the direction the parabola opens

Since the coefficient \( a \) is negative (\( a = -1 \)), the parabola opens downwards.

Step 3: Compare the width with \( y = x^2 \)

The coefficient \( |a| \) determines the width. Since \( |a| = 1 \), which is the same as for \( y = x^2 \), the parabola has the same shape.

Step 4: Find the vertex using the vertex formula

The vertex of a parabola in the form of \( ax^2 + bx + c \) is given by \[ x_v = \frac{-b}{2a} \] Substituting the coefficients, \[ x_v = \frac{-5}{2(-1)} = \frac{5}{2} \] Substitute \( x_v \) back into the equation to find \( y_v \):\[ y_v = -\left(\frac{5}{2}\right)^2 + 5\left(\frac{5}{2}\right) + 3 = -\frac{25}{4} + \frac{25}{2} + 3 =\frac{-25 + 50 + 12}{4} = \frac{37}{4} \] Thus, the vertex is \( \left( \frac{5}{2}, \frac{37}{4} \right) \).

Step 5: Calculate the discriminant

The discriminant for the quadratic equation is given by \[ D = b^2 - 4ac \] Substitute the coefficients: \[ D = 5^2 - 4(-1)(3) = 25 + 12 = 37 \] Since the discriminant is positive, the quadratic equation has 2 distinct real roots.

Step 6: Find the number of \( x \)-intercepts

Since the discriminant is positive (\( D = 37 \)), the graph intersects the \( x \)-axis at two distinct points, indicating 2 \( x \)-intercepts.

Key Concepts

Quadratic Equations

Quadratic equations are a fundamental part of algebra and are used to model various real-world scenarios. A standard quadratic equation is expressed in the form \(ax^2 + bx + c = 0\), where \(a, b,\) and \(c\) are constants and \(a eq 0\).
Key features of quadratic equations include:

• The highest exponent of the variable \(x\) is 2.
• It can open upwards or downwards based on the sign of the coefficient \(a\).
• The graph is a parabola.

The vertex of a quadratic equation, which is its highest or lowest point, can be calculated using the vertex formula \(x_v = \frac{-b}{2a}\). This helps in determining the maximum or minimum value of the quadratic expression.
By substituting \(x_v\) back into the original equation, you can find the y-coordinate of the vertex, denoted as \(y_v\). This gives the vertex \((x_v, y_v)\). Understanding quadratic equations is essential for solving many types of algebra problems and for graphing parabolas.

Discriminant

The discriminant is a key concept in quadratic equations and helps in determining the nature of the roots without solving the equation. For a quadratic equation \(ax^2 + bx + c = 0\), the discriminant \(D\) is given by \(D = b^2 - 4ac\).
The value of the discriminant provides insight into the roots of the equation:

• If \(D > 0\), the quadratic equation has 2 distinct real roots, meaning the parabola will intersect the x-axis at two points.
• If \(D = 0\), it indicates exactly one real root, or the parabola touches the x-axis at exactly one point (vertex).
• If \(D < 0\), there are no real roots, implying the parabola does not intersect the x-axis at any point.

Calculating the discriminant is crucial for understanding the behavior of quadratic equations in terms of their roots and can be a useful shortcut in solving problems quickly.

Parabola Direction

The direction in which a parabola opens is determined by the sign of the coefficient \(a\) in the quadratic equation \(ax^2 + bx + c\). Here's how it works:

• If \(a > 0\), the parabola opens upwards, forming a U-shape.
• If \(a < 0\), it opens downwards, forming an upside-down U-shape.

The direction is essential for understanding the vertex, the maximum or minimum point of the parabola. The coefficient \(a\) also influences the width of the parabola. If the absolute value of \(a\) is greater than 1, the parabola is narrower than the basic parabola \(y = x^2\). Conversely, if \(0 < |a| < 1\), it is wider. If \( |a| = 1\), it has the same width.
The direction and width together help in sketching the graph accurately and predicting the parabola's behavior. Understanding these properties of parabolas is fundamental in graphing quadratic equations and solving real-world problems modeled by such equations.