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Chapter 4: Problem 44

(a) find the vertex and the axis of symmetry of each quadratic function, and determine whether the graph is concave up or concave down. (b) Find the y-intercept and the \(x\) -intercepts, if any. (c) Use parts (a) and (b) to graph the function. (d) Find the domain and the range of the quadratic function. (e) Determine where the quadratic function is increasing and where it is decreasing. (f) Determine where \(f(x)>0\) and where \(f(x)<0\) \(f(x)=x^{2}-4 x\)

Short Answer

Expert verified
Vertex (2, -4); Axis of Symmetry \[ x = 2 \]; Concave Up; y-intercept (0,0); x-intercepts (0,0) and (4,0); Domain:\[ (-\infty, \infty) \]; Range:\[ [-4, \infty) \]; Increasing on\[ (2, \infty) \]; Decreasing on\[ (-\infty, 2) \]; \[ f(x) > 0 \text{ for } x < 0 \text{ or } x > 4 \]; \[ f(x) < 0 \text{ for } 0 < x < 4 \]

Step by step solution

01

- Find the Vertex

A quadratic function in the form of \[ f(x) = ax^2 + bx + c \] has its vertex at \[ x = -\frac{b}{2a} \]. For the given function \[ f(x) = x^2 - 4x \], we have \[ a = 1 \] and \[ b = -4 \]. Therefore, \[ x = -\frac{-4}{2 \cdot 1} = \frac{4}{2} = 2 \]. Substitute \[ x = 2 \] back into the function to find the y-coordinate of the vertex: \[ f(2) = 2^2 - 4(2) = 4 - 8 = -4 \]. Thus, the vertex is (2, -4).
02

- Determine the Axis of Symmetry

The axis of symmetry for a quadratic function is the vertical line that passes through the vertex. Given the vertex is at \[ x = 2 \], the axis of symmetry is the line \[ x = 2 \].
03

- Determine Concavity

Look at the leading coefficient \[ a \] of the quadratic function \[ f(x) = x^2 - 4x \]. Since \[ a = 1 \] (which is positive), the graph is concave up.
04

- Find the y-Intercept

The y-intercept occurs when \[ x = 0 \]. Substitute \[ x = 0 \] into the function: \[ f(0) = 0^2 - 4(0) = 0 \]. Thus, the y-intercept is (0, 0).
05

- Find the x-Intercepts

To find the x-intercepts, set \[ f(x) = 0 \] and solve for \[ x \]: \[ x^2 - 4x = 0 \]. Factoring gives \[ x(x - 4) = 0 \]. Solving this, we get \[ x = 0 \] or \[ x = 4 \]. Thus, the x-intercepts are (0, 0) and (4, 0).
06

- Graph the Function

Using the vertex \[ (2, -4) \], the y-intercept \[ (0, 0) \], and the x-intercepts \[ (0, 0) \] and \[ (4, 0) \], plot these points and draw the parabola opening upwards. The axis of symmetry is the vertical line \[ x = 2 \].
07

- Find the Domain and Range

The domain of any quadratic function is all real numbers \[ (-\infty, \infty) \]. The range is determined by the y-coordinate of the vertex and since the parabola opens upwards, the range is \[ [-4, \infty) \].
08

- Determine Increasing and Decreasing Intervals

The function is decreasing to the left of the vertex \[ x = 2 \] and increasing to the right of \[ x = 2 \]. Therefore, the function is decreasing on \[ (-\infty, 2) \] and increasing on \[ (2, \infty) \].
09

- Determine Where \( f(x) > 0 \) and \( f(x) < 0 \)

The function \[ f(x) \] is greater than zero when \[ x < 0 \] or \[ x > 4 \]. The function is less than zero in the interval \[ (0, 4) \].

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Key Concepts

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

vertex of a quadratic function
The vertex of a quadratic function represents the highest or lowest point of its graph, known as a parabola. For a quadratic function in standard form:

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Most popular questions from this chapter

A projectile is fired from a cliff 200 feet above the water at an inclination of \(45^{\circ}\) to the horizontal, with a muzzle velocity of 50 feet per second. The height \(h\) of the projectile above the water is modeled by $$ h(x)=\frac{-32 x^{2}}{50^{2}}+x+200 $$ where \(x\) is the horizontal distance of the projectile from the face of the cliff. (a) At what horizontal distance from the face of the cliff is the height of the projectile a maximum? (b) Find the maximum height of the projectile. (c) At what horizontal distance from the face of the cliff will the projectile strike the water? (d) Graph the function \(h, 0 \leq x \leq 200\). (e) Use a graphing utility to verify the solutions found in parts (b) and (c). (f) When the height of the projectile is 100 feet above the water, how far is it from the cliff?

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. If \(f(x)=\frac{2 x+B}{x-3}\) and \(f(5)=8,\) what is the value of \(B ?\)

Determine, without graphing, whether the given quadratic function has a maximum value or a minimum value, and then find the value. \(f(x)=-x^{2}+6 x-1\)

Show that the inequality \((x-4)^{2} \leq 0\) has exactly one solution.

(a) Graph fand g on the same Cartesian plane. (b) Solve \(f(x)=g(x)\) (c) Use the result of part (b) to label the points of intersection of the graphs of fand \(g\). (d) Shade the region for which \(f(x)>g(x)\); that is, the region below fand above \(g\). \(f(x)=2 x-1 ; \quad g(x)=x^{2}-4\)
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