Problem 11 - Algebra 1: Concepts and Skills

Question

Determine whether the graph of the function will intersect the x-axis in zero, one, or two points. \(y=-x^{2}-3 x+5\)

Step-by-Step Solution

Answer

The function \(y=-x^{2}-3x+5\) will intersect the x-axis at two points.

Step 1: Identify the coefficients

From the equation \( y=-x^{2}-3 x+5 \), the coefficients for \(x^{2}\), x, and the constant term are, respectively, -1 (a), -3 (b), and 5 (c).

Step 2: Calculate the discriminant

Substitute the coefficients into the discriminant formula \(b^{2} - 4ac\). So the discriminant D is \((-3)^{2}- 4*(-1)*5 = 9 - (-20) = 29.

Step 3: Determine the number of solutions

Since the discriminant D is greater than 0 (29>0), there will be two solutions to the equation, implying the function will intersect the x-axis at two points.

Key Concepts

Discriminant

The discriminant is a helpful mathematical tool when analyzing quadratic equations. It’s part of the quadratic formula, used to determine the nature and the number of roots of the quadratic equation, often in the shape of \( ax^2 + bx + c = 0 \). In simple terms, the discriminant (D) is calculated using the formula:

• \( D = b^2 - 4ac \)

What does it tell us? Well, it helps us understand how many times the parabola, which is the graph of the quadratic function, will touch or intersect the x-axis:

• If \( D > 0 \), the quadratic equation has two distinct real roots, meaning the parabola will intersect the x-axis at two different points.
• If \( D = 0 \), there is exactly one real root. The parabola touches the x-axis at only one point, referred to as a "double root."
• If \( D < 0 \), there are no real roots. The parabola does not intersect the x-axis at all; instead, it sits entirely above or below it.

In our exercise, since \( D = 29 > 0 \), the graph will intersect the x-axis at two points.

Graph of a function

A quadratic function graph typically appears as a parabola. Depending on the leading coefficient (the coefficient of \( x^2 \)), it can open upwards or downwards. This affects how the graph of the function intersects the x-axis.Let's break down the important elements:

• If the leading coefficient \( a > 0 \), the parabola opens upwards like a U-shape.
• If \( a < 0 \), as in our example \( a = -1 \), the parabola opens downwards like an upside-down U.

The vertex is a key feature of the graph, representing the highest or lowest point of the parabola depending on its orientation. It’s where the curve changes direction.The axis of symmetry is a vertical line that runs through the vertex, dividing the parabola into two mirror-image halves. This line can be found using the formula:

• \( x = -\frac{b}{2a} \)

For our function, understanding the shape and orientation of its graph lets us better predict how it interacts with the x-axis.

Intersecting the x-axis

In the context of quadratic functions, intersecting the x-axis means identifying where a function’s graph crosses or touches the x-axis. This intersection is significant because it provides the solutions, or "roots," of the quadratic equation. When we analyze the quadratic function \( y = -x^2 - 3x + 5 \), determining where the graph intersects the x-axis involves finding the values of \( x \) for which \( y = 0 \). This is solved by setting the quadratic equation \( 0 = -x^2 - 3x + 5 \) and finding the roots. Coinciding with our earlier discussion on the discriminant, if \( D > 0 \), we expect two distinct real roots. Each root represents one point where the graph intersects the x-axis. For our function with \( D = 29 \), there are indeed two intersection points. Understanding this helps you grasp why the discriminant is a powerful predictor and how it ties directly into the graph’s behavior in relation to the x-axis.