Problem 42 - Algebra for College Students

Question

Find all intercepts for the graph of each quadratic function. $$f(x)=-2 x^{2}-x+3$$

Step-by-Step Solution

Answer

The y-intercept is (0, 3). The x-intercepts are \((-\frac{3}{2}, 0)\) and \((1, 0)\).

Step 1: Identify the y-intercept

To find the y-intercept, set x to 0 and solve for f(0). For the function: \(f(x) = -2x^2 - x + 3\), let x = 0:\(f(0) = -2(0)^2 - (0) + 3 = 3\). The y-intercept is at (0, 3).

Step 2: Find the x-intercepts

To find the x-intercepts, set f(x) to 0 and solve for x. For the function: \(f(x) = -2x^2 - x + 3\), set f(x) to 0:\(0 = -2x^2 - x + 3\). Solve the quadratic equation using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = -2\), \(b = -1\), and \(c = 3\).

Step 3: Calculate the discriminant

Calculate the discriminant to determine the number of x-intercepts. The discriminant is given by \(b^2 - 4ac\). For our equation:\(b = -1\), \(a = -2\), \(c = 3\).So the discriminant is:\((-1)^2 - 4(-2)(3) = 1 + 24 = 25\).

Step 4: Solve for x-intercepts

Since the discriminant is positive, there are two real solutions. Use the quadratic formula:\(x = \frac{-(-1) \pm \sqrt{25}}{2(-2)}\) Simplify the expression:\(x = \frac{1 \pm 5}{-4}\). So the solutions are:\(x = \frac{1 + 5}{-4} = -\frac{6}{4} = -\frac{3}{2}\), and\(x = \frac{1 - 5}{-4} = \frac{-4}{-4} = 1\). The x-intercepts are at \((-\frac{3}{2}, 0)\) and \((1, 0)\).

Key Concepts

quadratic equations

Quadratic equations are polynomial equations of degree two. They have the general form:
ax^2 + bx + c = 0, where:

• a, b, and c are coefficients
• x is the variable

Each quadratic equation graph forms a parabola. The parabola could open upwards or downwards depending on the sign of the coefficient a. If a is positive, the parabola opens upwards. If a is negative, it opens downwards.

x-intercepts

The x-intercepts of a quadratic function are points where the graph crosses the x-axis. To find the x-intercepts, set the quadratic function to zero and solve for x.
In our example, for the function: -2x^2 - x + 3 = 0, first, we apply the quadratic formula to find the x-values where the function crosses the x-axis.

• First, calculate the discriminant to ensure real solutions exist.

If the discriminant is positive, there are two real x-intercepts. If it is zero, there is one real x-intercept. If negative, there are no real x-intercepts.

y-intercepts

The y-intercept is the point where the graph crosses the y-axis. To find it, set x to 0 in the quadratic function and solve for f(0).
For our function: f(x) = -2x^2 - x + 3,

• Let x = 0: f(0) = -2(0)^2 - (0) + 3 = 3

Therefore, the y-intercept is at (0, 3). This is the point where the graph intersects the y-axis.

quadratic formula

The quadratic formula is used for solving quadratic equations:

Quadratic Formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

• a, b, and c are the coefficients of the quadratic equation
• \(b^2 - 4ac\) is the discriminant

In our example:

Given:

\[ a = -2, b = -1, c = 3. \]

Plugging values into the formula gives:

\[ x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(-2)(3)}}{2(-2)} \]
Simplify to find the roots where the graph crosses the x-axis.

discriminant

The discriminant helps determine the nature of the roots for a quadratic equation. It is the part of the quadratic formula under the square root:
\[ b^2 - 4ac \]

• If the discriminant is positive (>0), the equation has two distinct real roots.
• If it is zero, there is exactly one real root.
• If negative (<0), there are no real roots, only complex ones.

In our example, \[- 2x^2 - x + 3\]

Given:

b = -1, a = -2, c = 3.

Calculate:

\[ (-1)^2 - 4(-2)(3) = 1 + 24 = 25 \]
With a positive discriminant of 25, the quadratic equation has two real and distinct x-intercepts.