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Magazine Chapter 10: Problem 60
Find the \(x\) - and \(y\) -intercepts of the graph of the quadratic function. See Example \(9 .\) $$ f(x)=-x^{2}-10 x-21 $$
Short Answer
Expert verified
The y-intercept is (0, -21) and the x-intercepts are (-7, 0) and (-3, 0).
Step by step solution
01
Find the y-intercept
To find the y-intercept of the function, we substitute the value of 0 for \(x\) in the function \(f(x)\). So we compute \(f(0) = -0^2 - 10 \times 0 - 21 = -21\). Thus, the y-intercept is \( (0, -21) \).
02
Find the x-intercepts
To find the x-intercepts of the function, we need to solve the equation \(-x^2 - 10x - 21 = 0\). This is a quadratic equation of the form \(ax^2 + bx + c = 0\). The best approach here is to use the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
03
Determine the coefficients
Identify the coefficients of the quadratic equation \(-x^2 - 10x - 21 = 0\). Here, \(a = -1\), \(b = -10\), and \(c = -21\).
04
Calculate the discriminant
The discriminant of the quadratic equation is given by \(b^2 - 4ac\). Substitute the values: \((-10)^2 - 4(-1)(-21) = 100 - 84 = 16\). The discriminant is positive (16), indicating two real roots.
05
Solve for x-intercepts using the quadratic formula
Substitute the values into the quadratic formula: \[x = \frac{-(-10) \pm \sqrt{16}}{2(-1)}\]\[x = \frac{10 \pm 4}{-2}\]This gives us two solutions:1. \(x = \frac{10 + 4}{-2} = -7\)2. \(x = \frac{10 - 4}{-2} = -3\)Thus, the x-intercepts are \((-7, 0)\) and \((-3, 0)\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
X-Intercepts
X-intercepts are the points where a graph crosses the x-axis. At these points, the value of the function is zero. In other words, when you seek the x-intercepts of a function, you're looking for the values of x for which the function equals zero.
To find these, you can set the quadratic equation to zero. For a quadratic function like \(-x^2 - 10x - 21\), the goal is to solve \(-x^2 - 10x - 21 = 0\).
The x-intercepts of this equation can be calculated using the quadratic formula, which we'll talk more about later. Once computed, the x-intercepts are expressed as ordered pairs like \((-7, 0)\) and \((-3, 0)\), indicating where the curve cuts the x-axis.
These points are essential because they show the solutions to the function and help in understanding the shape and position of the parabola.
To find these, you can set the quadratic equation to zero. For a quadratic function like \(-x^2 - 10x - 21\), the goal is to solve \(-x^2 - 10x - 21 = 0\).
The x-intercepts of this equation can be calculated using the quadratic formula, which we'll talk more about later. Once computed, the x-intercepts are expressed as ordered pairs like \((-7, 0)\) and \((-3, 0)\), indicating where the curve cuts the x-axis.
These points are essential because they show the solutions to the function and help in understanding the shape and position of the parabola.
Y-Intercepts
Y-intercepts occur where the graph crosses the y-axis. At this point, the value of x is zero. Therefore, to discover the y-intercept of a quadratic function, you substitute zero for x in the equation.
For the function \(f(x) = -x^2 - 10x - 21\), you calculate \(f(0)\). This results in the expression \(-0^2 - 10 \times 0 - 21 = -21\).
Thus, the y-intercept is the point \((0, -21)\).
The y-intercept serves as a starting point and gives insight into where the curve begins on the vertical axis. It's crucial in giving us a sense of the whole graph's location relative to the origin.
For the function \(f(x) = -x^2 - 10x - 21\), you calculate \(f(0)\). This results in the expression \(-0^2 - 10 \times 0 - 21 = -21\).
Thus, the y-intercept is the point \((0, -21)\).
The y-intercept serves as a starting point and gives insight into where the curve begins on the vertical axis. It's crucial in giving us a sense of the whole graph's location relative to the origin.
Quadratic Formula
The quadratic formula is a powerful tool used to find the roots of any quadratic equation in the form \(ax^2 + bx + c = 0\). This formula is:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
Using this formula, you can determine the x-intercepts of a function. For the function \(-x^2 - 10x - 21\), we identify \(a = -1\), \(b = -10\), and \(c = -21\).
After substituting these values into the quadratic formula, you solve for x. Here, the solutions were \(x = -7\) and \(x = -3\), which are the x-intercepts.
Using the quadratic formula is very useful as it provides a direct method to solve complex quadratic equations without guessing, ensuring you find accurate results every time.
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
Using this formula, you can determine the x-intercepts of a function. For the function \(-x^2 - 10x - 21\), we identify \(a = -1\), \(b = -10\), and \(c = -21\).
After substituting these values into the quadratic formula, you solve for x. Here, the solutions were \(x = -7\) and \(x = -3\), which are the x-intercepts.
Using the quadratic formula is very useful as it provides a direct method to solve complex quadratic equations without guessing, ensuring you find accurate results every time.
Discriminant
The discriminant is a part of the quadratic formula and is found under the square root in the formula: \(b^2 - 4ac\). It gives us useful information about the nature of the roots of the quadratic equation.
For \(-x^2 - 10x - 21\), we calculated the discriminant as \(16\). Because it's positive, the equation has two real roots, which we found to be \(-7\) and \(-3\).
This part of the quadratic form helps in understanding the number and type of solutions without solving the entire equation, making it a crucial concept in analyzing quadratic functions.
- If the discriminant is positive, there are two distinct real roots.
- If it is zero, there is exactly one real root (also called a repeated root).
- If it is negative, there are no real roots, only complex ones.
For \(-x^2 - 10x - 21\), we calculated the discriminant as \(16\). Because it's positive, the equation has two real roots, which we found to be \(-7\) and \(-3\).
This part of the quadratic form helps in understanding the number and type of solutions without solving the entire equation, making it a crucial concept in analyzing quadratic functions.
