Problem 42 - Algebra and Trigonometry

Question

\(41-48\) Find the \(x\) - and \(y\) -intercepts of the graph of the equation. $$ y=x^{2}-5 x+6 $$

Step-by-Step Solution

Answer

Y-intercept: (0, 6); X-intercepts: (2, 0) and (3, 0).

Step 1: Identify the Intercepts

The x-intercept is where the graph of the equation crosses the x-axis. For this point, the y-value is 0. For the y-intercept, it's where the graph crosses the y-axis, which happens when the x-value is 0.

Step 2: Find the Y-intercept

To find the y-intercept, set \(x = 0\) in the equation \(y = x^2 - 5x + 6\) and solve for \(y\):\[y = 0^2 - 5(0) + 6 = 6\]This means the y-intercept is 6, and the point is (0, 6).

Step 3: Set Up the X-intercept Equation

To find the x-intercept, set \(y = 0\) in the equation and solve for \(x\). The equation becomes:\[0 = x^2 - 5x + 6\]

Step 4: Solve the Quadratic Equation

To solve \(x^2 - 5x + 6 = 0\), factor the quadratic equation:\[(x - 2)(x - 3) = 0\]Set each factor equal to zero and solve for \(x\):\[x - 2 = 0 \quad \Rightarrow \quad x = 2\]\[x - 3 = 0 \quad \Rightarrow \quad x = 3\]

Step 5: State the X-intercepts

The solutions \(x = 2\) and \(x = 3\) are the x-intercepts. Thus, the points are (2, 0) and (3, 0).

Key Concepts

Quadratic Equations

Quadratic equations are equations of the form \(y = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). These equations represent parabolas when graphed on the coordinate plane.
A quadratic equation like \(y = x^2 - 5x + 6\) features some distinctive properties:

• It always features an \(x^2\) term, making the graph a parabola.
• The parabola can open upwards or downwards depending on the sign of \(a\). In this case, \(a = 1\), and the parabola opens upwards.
• The vertex of this parabola provides a turning point, but intercepts particularly help understand where the graph crosses the axes.

Understanding the structure of quadratic equations helps in identifying patterns regarding roots, vertex, and intercepts.
The solutions to the equation, also known as the roots, can be found using factoring, completing the square, or the quadratic formula, depending on what is most convenient for the given equation.

X-intercept

The x-intercepts of a graph are the points where the graph crosses the x-axis. These points occur where the value of \(y\) is zero. To find x-intercepts in a quadratic equation like \(y = x^2 - 5x + 6\), we set the equation to 0 and solve for \(x\):

• Start by substituting \(y = 0\) to form the equation \(0 = x^2 - 5x + 6\).
• Factor the quadratic equation, if possible. In this case, it factors to \((x - 2)(x - 3) = 0\).
• Use the Zero Product Property: set each factor equal to zero and solve, giving us \(x - 2 = 0\) or \(x - 3 = 0\).
• Solving these gives the solutions \(x = 2\) and \(x = 3\).

Thus, the x-intercepts are the points \((2, 0)\) and \((3, 0)\), signifying the locations where the parabola crosses the x-axis. Understanding x-intercepts is vital for solving quadratic equations and analyzing the parabolic movement across axes in graphs.

Y-intercept

The y-intercept of a graph is the point where the graph crosses the y-axis. This happens when the value of \(x\) is zero. To find the y-intercept in the given quadratic equation \(y = x^2 - 5x + 6\), we substitute \(x = 0\) in the equation and solve for \(y\):

• Replace \(x\) with zero: \(y = 0^2 - 5(0) + 6\).
• Simplify the equation to find \(y = 6\).

This calculation indicates that the y-intercept is the point \((0, 6)\).
Graphically, this means this point is where the graph intercepts the y-axis.
Knowing the y-intercept provides insight into the starting point of the graph on the vertical axis, which helps in sketching the parabola's general shape and trajectory on the graph.