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Chapter 8: Problem 16

Identify the \(x\) -intercepts of the following functions; then graph the functions to check your work. a. \(y=3 x+6\) b. \(y=(x+4)(x-1)\) c. \(y=(x+5)(x-3)(2 x+5)\)

Short Answer

Expert verified
The x-intercepts are: a. \(x = -2\), b. \(x = -4\) and \(x = 1\), c. \(x = -5\), \(x = 3\), and \(x = -\frac{5}{2}\).

Step by step solution

01

- Identify the x-intercepts for y=3x+6

To find the x-intercepts, set y to 0 and solve for x: \(0 = 3x + 6\).
02

- Solve for x in y=3x+6

Solve \(3x + 6 = 0\) by subtracting 6 from both sides: \(3x = -6\). Then, divide both sides by 3: \(x = -2\). Hence, the x-intercept is \(x = -2\).
03

- Identify the x-intercepts for y=(x+4)(x-1)

Set y to 0: \(0 = (x + 4)(x - 1)\). Use the zero-product property to set each factor to 0: \(x + 4 = 0\) or \(x - 1 = 0\).
04

- Solve for x in y=(x+4)(x-1)

Solve \(x + 4 = 0\) giving \(x = -4\). Solve \(x - 1 = 0\) giving \(x = 1\). Hence, the x-intercepts are \(x = -4\) and \(x = 1\).
05

- Identify the x-intercepts for y=(x+5)(x-3)(2x+5)

Set y to 0: \(0 = (x + 5)(x - 3)(2x + 5)\). Use the zero-product property to set each factor to 0: \(x + 5 = 0\), \(x - 3 = 0\), or \(2x + 5 = 0\).
06

- Solve for x in y=(x+5)(x-3)(2x+5)

Solve \(x + 5 = 0\) giving \(x = -5\). Solve \(x - 3 = 0\) giving \(x = 3\). Solve \(2x + 5 = 0\) giving \(2x = -5\) and \(x = -\frac{5}{2}\). Hence, the x-intercepts are \(x = -5\), \(x = 3\), and \(x = -\frac{5}{2}\).
07

- Check solutions by graphing the functions

Graph each function and verify that the x-intercepts match the solutions obtained: For \(y = 3x + 6\), the graph intersects the x-axis at \(x = -2\). For \(y = (x + 4)(x - 1)\), the graph intersects the x-axis at \(x = -4\) and \(x = 1\). For \(y = (x + 5)(x - 3)(2x + 5)\), the graph intersects the x-axis at \(x = -5\), \(x = 3\), and \(x = -\frac{5}{2}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

linear functions
Linear functions are the simplest type of functions and can be written in the form of \[y = ax + b\]. In these functions, 鈥榓鈥� and 鈥榖鈥� are constants. The graph of a linear function is always a straight line.

To find the x-intercept of a linear function, you need to set y to 0 and solve for x. For example, in the equation \[y = 3x + 6\], you set y to 0 to get: \[0 = 3x + 6\]. Solving this, you find that \[x = -2\]. This means the x-intercept of the function is at \[x = -2\].
quadratic functions
Quadratic functions are functions of the form \[y = ax^2 + bx + c\], where 'a', 'b', and 'c' are constants. The graph of a quadratic function is a parabola, which can open upwards or downwards.

To find the x-intercepts of a quadratic function, you set the function equal to zero and solve for x. Consider the function \[y = (x + 4)(x - 1)\]. To find the x-intercepts, you set y to 0 and solve the equation: \[0 = (x + 4)(x - 1)\]. Using the zero-product property, you set each factor to zero: \[x + 4 = 0\] and \[x - 1 = 0\]. Solving these, you get \[x = -4\] and \[x = 1\]. So, the x-intercepts are -4 and 1.
polynomial functions
Polynomial functions are functions that can be written as the sum of terms consisting of a variable raised to a non-negative integer power multiplied by a coefficient. An example of a polynomial function is \[y = (x + 5)(x - 3)(2x + 5)\].

To find the x-intercepts of this function, you set y to 0 and solve: \[0 = (x + 5)(x - 3)(2x + 5)\]. Using the zero-product property, you set each factor to zero: \[x + 5 = 0\], \[x - 3 = 0\], and \[2x + 5 = 0\]. This gives you the solutions \[x = -5\], \[x = 3\], and \[x = -\frac{5}{2}\]. These are the x-intercepts of the polynomial function.
graphing functions
Graphing functions is a useful way to visualize them and their intercepts.

When graphing linear, quadratic, or polynomial functions, the x-intercepts are where the graph crosses the x-axis. These are the points where the function equals zero. For example:
  • Graphing \[y = 3x + 6\], the graph crosses the x-axis at \[x = -2\].
  • Graphing \[y = (x + 4)(x - 1)\], the graph crosses the x-axis at \[x = -4\] and \[x = 1\].
  • Graphing \[y = (x + 5)(x - 3)(2x + 5)\], the graph crosses the x-axis at \[x = -5\], \[x = 3\], and \[x = -\frac{5}{2}\].
Graphing these can help verify your solutions.
zero-product property
The zero-product property is a fundamental concept in algebra that states: if the product of two or more factors is zero, then at least one of the factors must be zero.

When solving for x-intercepts, this rule is very helpful. For example, consider the equation \[0 = (x + 5)(x - 3)(2x + 5)\]. Using the zero-product property, set each individual factor to zero:\[x + 5 = 0\], \[x - 3 = 0\], and \[2x + 5 = 0\]. This gives the solutions \[x = -5\], \[x = 3\], and \[x = -\frac{5}{2}\]. These values make each factor zero, confirming the x-intercepts.

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Most popular questions from this chapter

Describe the behavior of each polynomial function for large values (positive or negative) of the independent variable and estimate the maximum number of turning points. If available, use technology to verify the actual number. a. \(y=-2 x^{4}+4 x+3\) b. \(y=\left(t^{2}+1\right)\left(t^{2}-1\right)\) c. \(y=x^{3}+x+1\) d. \(y=x^{5}-3 x^{4}-11 x^{3}+3 x^{2}+10 x\)

Put each of the following quadratics into standard form. a. \(g(x)=(2 x-1)(x+3)\) b. \(h(r)=(5 r+2)(2 r+5)\) c. \(R(t)=(5-2 t)(3-4 t)\)

Solve using the quadratic formula. a. \(x^{2}-3 x=12\) e. \(\frac{1}{x-2}=\frac{x+1}{x-1}\) b. \(3 x^{2}=4 x+2\) f. \(\frac{x^{2}}{3}+\frac{x}{2}-\frac{1}{6}=0\) c. \(3\left(x^{2}+1\right)=x+2\) g. \(\frac{1}{x^{2}}-\frac{3}{x}=\frac{1}{6}\) d. \((3 x-1)(x+2)=4\)

A pilot has crashed in the Sahara Desert. She still has her maps and knows her position, but her radio is destroyed. Her only hope for rescue is to hike out to a highway that passes near her position. She needs to determine the closest point on the highway and how far away it is. a. The highway is a straight line passing through a point 15 miles due north of her and another point 20 miles due east. Draw a sketch of the situation on graph paper, placing the pilot at the origin and labeling the two points on the highway. b. Construct an equation that represents the highway (using \(x\) for miles east and \(y\) for miles north). c. Now use the Pythagorean Theorem to describe the square of the distance, \(d,\) of the pilot to any point \((x, y)\) on the highway. d. Substitute the expression for \(y\) from part (b) into the equation from part (c) in order to write \(d^{2}\) as a quadratic in \(x\) e. If we minimize \(d^{2}\), we minimize the distance \(d\). So let \(D=d^{2}\) and write \(D\) as a quadratic function in \(x\). Now find the minimum value for \(D\). f. What are the coordinates of the closest point on the highway, and what is the distance, \(d\), to that point?

Salt is applied to roads to decrease the temperature at which icing occurs, Assume that with no salt, icing occurs at \(32^{\circ} \mathrm{F}\), and that each unit increase in the density of salt applied decreases the icing temperature by \(5^{\circ} \mathrm{F}\). a. Construct a formula for icing temperature, \(T\), as a function of salt density, \(s\). Trucks spread salt on the road, but they do not necessarily spread it uniformly across the road surface. If the edges of the road get half as much salt as the middle, we can describe salt density \(S(x)\) as a function of the distance, \(x\), from the center of the road by \(S(x)=\left[1-\frac{1}{2}\left(\frac{x}{k}\right)^{2}\right] S_{d},\) where \(k\) is the distance from the centerline to the road edges and \(S_{d}\) is the salt density applied in the middle of the road. b. What will the expression for \(S(x)\) be if the road is 40 feet wide? c. What will the value for \(x\) be at the middle of the 40 -footwide road? At the edge of the road? Verify that at the middle of the road the value of the salt density \(S(x)\) is \(S_{d}\) and that at the edge the value of \(S(x)\) is \(\frac{1}{2} S_{d}\) d. Construct a function that describes the icing temperature, \(T,\) as a function of \(x,\) the distance from the center of the 40 -foot-wide road. e. What is the icing temperature at the middle of the 40 -foot-wide road? At the edge?
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