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Chapter 3: Problem 58

Determine the number of \(x\) -intercepts of the graph of \(f(x)=a x^{2}+b x+c(a \neq 0),\) based on the discriminant of the related equation \(f(x)=0\). (Hint: Recall that the discriminant is \(\left.b^{2}-4 a c .\right)\) $$ f(x)=-2 x^{2}+5 x-10 $$

Short Answer

Expert verified
The graph has no real \( x \)-intercepts.

Step by step solution

01

Identify the coefficients

Identify the coefficients in the given quadratic function \( f(x) = -2x^2 + 5x - 10 \). Here, \( a = -2 \), \( b = 5 \), and \( c = -10 \).
02

Recall the discriminant formula

The discriminant \( \triangle \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by \( b^2 - 4ac \).
03

Substitute the coefficients into the discriminant

Substitute \( a = -2 \), \( b = 5 \), and \( c = -10 \) into the discriminant formula: \[ \triangle = 5^2 - 4(-2)(-10) \].
04

Simplify the discriminant

Simplify the expression to find the value of the discriminant: \[ \triangle = 25 - 80 = -55 \].
05

Determine the number of x-intercepts

The number of \( x \)-intercepts depends on the value of the discriminant \( \triangle \):- If \( \triangle > 0 \), there are 2 distinct real \( x \)-intercepts.- If \( \triangle = 0 \), there is exactly 1 real \( x \)-intercept.- If \( \triangle < 0 \), there are no real \( x \)-intercepts.Since \( \triangle = -55 < 0 \), the graph has no real \( x \)-intercepts.

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Key Concepts

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

Quadratic Function
A quadratic function is a type of polynomial function represented by the general form: \(f(x) = ax^2 + bx + c\), where \(a eq 0\). This function forms a parabola when graphed on a coordinate plane. The parameter \(a\) determines the direction of the parabola (opening upwards if \(a > 0\) and downwards if \(a < 0\)), while \(b\) and \(c\) adjust the slope and position of the parabola. Understanding the quadratic function is key to solving many problems in algebra and calculus.
X-Intercepts
The \(x\)-intercepts of a quadratic function are the points where the graph crosses the x-axis. To find these intercepts, set \(f(x) = 0\) and solve the corresponding quadratic equation. The solutions to this equation give the \(x\)-coordinates of these intercepts. The number of solutions (or \(x\)-intercepts) is determined by the discriminant \(\Delta\). Therefore, knowing the discriminant helps us quickly identify how many times a quadratic graph intersects the x-axis:
  • If \(\Delta > 0\), there are 2 distinct real \(x\)-intercepts.
  • If \(\Delta = 0\), there is exactly 1 real \(x\)-intercept.
  • If \(\Delta < 0\), there are no real \(x\)-intercepts.
This is important for visualizing the behavior of quadratic functions.
Coefficients
In the quadratic function \(f(x) = ax^2 + bx + c\), \(a\), \(b\), and \(c\) are known as the coefficients. Each coefficient serves a distinct role:
  • \(a\) (the leading coefficient) influences the direction and width of the parabola.
  • \(b\) affects the location of the vertex horizontally.
  • \(c\) is the constant term and indicates the y-intercept of the function (the point where the parabola crosses the y-axis).
By identifying and analyzing these coefficients, we can determine key characteristics of the quadratic function and its graph.
Discriminant Formula
The discriminant of a quadratic equation \(ax^2 + bx + c = 0\) is given by the formula \(\Delta = b^2 - 4ac\). This value determines the nature of the roots of the equation, which in turn affects the number of \(x\)-intercepts of the quadratic function. In our given function \(f(x) = -2x^2 + 5x - 10\), we identified the coefficients as \(a = -2\), \(b = 5\), and \(c = -10\).
Using the discriminant formula:
\[\Delta = 5^2 - 4(-2)(-10) = 25 - 80 = -55\]
Since the discriminant is negative \((\Delta < 0)\), it indicates that the quadratic equation has no real roots. Consequently, the graph of the quadratic function does not intersect the x-axis, hence there are no real \(x\)-intercepts.

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Most popular questions from this chapter

a. Factor the polynomial over the set of real numbers. b. Factor the polynomial over the set of complex numbers. $$f(x)=x^{4}+2 x^{2}-35$$

The procedure to solve a polynomial or rational inequality may be applied to all inequalities of the form \(f(x)>0, f(x)<0,\) \(f(x) \geq 0,\) and \(f(x) \leq 0 .\) That is, find the real solutions to the related equation and determine restricted values of \(x .\) Then determine the sign of \(f(x)\) on each interval defined by the boundary points. Use this process to solve the inequalities. $$ \sqrt{4-x}-6 \geq 0 $$

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