Problem 40 - Algebra for College Students

Question

Find all intercepts for the graph of each quadratic function. $$y=x^{2}-x-6$$

Step-by-Step Solution

Answer

The y-intercept is (0, -6). The x-intercepts are (3, 0) and (-2, 0).

Step 1: Find the y-intercept

To find the y-intercept of the graph, set x to 0 in the equation and solve for y. \[ y = 0^2 - 0 - 6 \] \[ y = -6 \] The y-intercept is (0, -6).

Step 2: Find the x-intercepts by factoring

To find the x-intercepts, set y to 0 in the equation and solve for x. \[ 0 = x^2 - x - 6 \] Factor the quadratic equation: \[ 0 = (x - 3)(x + 2) \] Set each factor to zero: \[ x - 3 = 0 \text{ or } x + 2 = 0 \] Solve for x: \[ x = 3 \] \[ x = -2 \] The x-intercepts are (3, 0) and (-2, 0).

Key Concepts

y-intercept

To find the y-intercept of a quadratic function, you simply need to set the x-value to zero and solve for y. This is because the y-intercept is the point where the graph crosses the y-axis, which is always at x = 0.

Let's go through an example with the equation: \( y = x^2 - x - 6 \).

Setting x to 0 gives us: \[ y = 0^2 - 0 - 6 \] \[ y = -6 \]

Therefore, the y-intercept for this quadratic function is \( (0, -6) \). This tells us that the graph will cross the y-axis at the point where y is -6.

x-intercepts

Finding x-intercepts involves setting y to zero and solving for x, as these are the points where the graph crosses the x-axis (y = 0).

Using the same equation from earlier: \( y = x^2 - x - 6 \), we set y to 0:

\[ 0 = x^2 - x - 6 \]

This equation needs to be solved for x to find the intercepts. The x-intercepts of this function are the points where the graph cuts the x-axis.

factoring quadratics

To solve the quadratic equation, we use a method called factoring. Factoring involves breaking down the quadratic equation into simpler expressions (factors) that can be multiplied together to return to the original equation.

For our equation \( x^2 - x - 6 \), we look for two numbers that multiply to -6 (the constant term) but add up to -1 (the coefficient of the middle term).

The pair of numbers -3 and 2 fit these criteria, so we can write:

\[ x^2 - x - 6 = (x - 3)(x + 2) \]

This means our quadratic equation can be rewritten as \( (x - 3)(x + 2) = 0 \).

solving equations

Now, we can solve the equation by setting each factor to zero. This is based on the zero-product property, which states that if a product of factors equals zero, at least one of the factors must be zero.

So, we solve the following equations:

\[ x - 3 = 0 \] \[ x + 2 = 0 \]

Solving these, we get:

\[ x = 3 \] \[ x = -2 \]

Therefore, the x-intercepts of the quadratic function \( y = x^2 - x - 6 \) are \( (3, 0) \) and \( (-2, 0) \). These points indicate where the graph crosses the x-axis.