Problem 60 - Algebra 2

Question

Without graphing, tell how many \(x\) -intercepts each function has. $$ y=-x^{2}+3 x+10 $$

Step-by-Step Solution

Answer

The function has two x-intercepts.

Step 1: Identify the Type of Function

The given function is a quadratic, which is represented by the general form of a polynomial of degree 2: \(y=ax^2+bx+c\). The number of x-intercepts of a quadratic function can be determined by finding the roots of the equation.

Step 2: Find the Discriminant

The discriminant of a quadratic equation \(ax^2+bx+c=0\) can be found using the formula \(D=b^2-4ac\). The discriminant helps in determining the nature of roots. If \(D>0\), there are two distinct real roots (x-intercepts); if \(D=0\), there is exactly one real root (a repeated x-intercept); if \(D<0\), there are no real roots (no x-intercepts).

Step 3: Calculate the Discriminant for the Given Function

For the function \(y=-x^2+3x+10\), we identify \(a=-1\), \(b=3\), and \(c=10\). Now we calculate the discriminant \(D=b^2-4ac=(3)^2-(4)(-1)(10)=9+40=49\).

Step 4: Determine the Number of x-intercepts

Since the discriminant \(D=49\) is greater than 0, the quadratic equation has two distinct real roots. This means that the graph of the function will cross the x-axis at two points, so the function has two x-intercepts.

Key Concepts

Discriminant of Quadratic Equations

The discriminant of a quadratic equation holds the key to understanding the nature of its roots. Given a quadratic equation in the standard form, which is represented as \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are coefficients, the discriminant, symbolized by \( D \), can be calculated using the formula \( D = b^2 - 4ac \).

This value allows us to make precise predictions regarding the roots without actually solving the equation:

• If \( D > 0 \), the quadratic has two distinct real roots.
• If \( D = 0 \), the equation has one real root, also called a repeated or double root.
• If \( D < 0 \), the equation has complex roots, which are not real numbers.

When understanding x-intercepts on the graph of a quadratic function, the discriminant indicates how many times the parabola will touch or cross the x-axis. To further clarify how this concept applies, we'll look at an example.

Roots of Quadratic Equations

The roots, or solutions, of a quadratic equation occur where the graph intersects the x-axis. These points are referred to as x-intercepts or zeros of the function. To find the roots, we can use various methods such as factoring, completing the square, or using the quadratic formula: \( x = \frac{{-b \pm \sqrt{{D}}}}{{2a}} \).

The roots are dependent on the value of the discriminant. For our example, where the discriminant is 49, the quadratic formula tells us there are two real roots. This also correlates with the fact that a graph of a quadratic equation with real roots will intersect the x-axis at these points. Visually, each root on the graph is a point where the parabola changes direction from either going downwards to upwards or vice versa. Finding the roots is essential not just for graphing but also for understanding the function's behavior.

Nature of Quadratic Roots

The nature of the roots of a quadratic equation describes whether the solutions are real or complex and whether they are distinct or repeated. Based on the discriminant \( D \), we can categorize the roots as follows:

• Real and Distinct: If \( D > 0 \), the roots are real and distinct, meaning the graph of the quadratic function will intersect the x-axis at two distinct points.
• Real and Repeated: If \( D = 0 \), there is one real root that the function touches the x-axis at, which is also the vertex of the parabola.
• Complex: If \( D < 0 \), the roots are complex and the parabola does not intersect the x-axis at all.

For the function \( y = -x^2 + 3x + 10 \), since \( D = 49 > 0 \), it indicates that this function has two real and distinct roots. This directly translates to there being two x-intercepts for the graph, where the shape of the graph, a downward-opening parabola, will cross the x-axis twice, confirming the existence of two distinct solutions to the equation.