Problem 15
Question
For the following exercises, determine whether there is a minimum or maximum value to each quadratic function. Find the value and the axis of symmetry. $$ f(x)=2 x^{2}-10 x+4 $$
Step-by-Step Solution
Verified Answer
The function has a minimum value of -8.5 at the axis of symmetry \( x = 2.5 \).
1Step 1: Identify the Quadratic Function
The given quadratic function is \( f(x) = 2x^2 - 10x + 4 \). Here, the coefficient \( a = 2 \), \( b = -10 \), and \( c = 4 \).
2Step 2: Determine the Direction of the Parabola
Since the coefficient \( a = 2 \) is positive, the parabola opens upwards, which means there is a minimum value.
3Step 3: Find the Axis of Symmetry
The axis of symmetry for a quadratic function \( ax^2 + bx + c \) is given by the formula \( x = -\frac{b}{2a} \). Substituting the given values, \( x = -\frac{-10}{2 \times 2} = \frac{10}{4} = 2.5 \).
4Step 4: Calculate the Minimum Value
To find the minimum value, substitute \( x = 2.5 \) into the function: \( f(2.5) = 2(2.5)^2 - 10(2.5) + 4 \). First, find \( (2.5)^2 = 6.25 \), then calculate \( 2 \times 6.25 = 12.5 \). Next, \( 10 \times 2.5 = 25 \). Finally, calculate \( 12.5 - 25 + 4 = -8.5 \). So, the minimum value is \( -8.5 \).
Key Concepts
ParabolaAxis of SymmetryMinimum ValueQuadratic Formula
Parabola
A parabola is a U-shaped graph that represents a quadratic function. It can either open upwards or downwards. The direction in which a parabola opens is determined by the coefficient of the squared term in the quadratic function, often denoted as "a."
If \( a > 0 \), the parabola opens upwards, forming a happy face shape. If \( a < 0 \), it opens downwards, forming a sad face. In our example, the quadratic function \( f(x) = 2x^2 - 10x + 4 \) has \( a = 2 \), which means it opens upwards. This is crucial because it tells us whether the quadratic has a minimum or maximum value. Since the parabola opens upwards, the vertex represents the minimum point of the function.
If \( a > 0 \), the parabola opens upwards, forming a happy face shape. If \( a < 0 \), it opens downwards, forming a sad face. In our example, the quadratic function \( f(x) = 2x^2 - 10x + 4 \) has \( a = 2 \), which means it opens upwards. This is crucial because it tells us whether the quadratic has a minimum or maximum value. Since the parabola opens upwards, the vertex represents the minimum point of the function.
Axis of Symmetry
The axis of symmetry is an imaginary vertical line that divides a parabola into two mirror-image halves. It helps in easily finding the vertex of the parabola, which is either the minimum or maximum point of the graph.
Mathematically, the axis of symmetry for a quadratic equation \( ax^2 + bx + c \) is given by the formula \( x = -\frac{b}{2a} \). This gives us the x-coordinate of the vertex. In the given function \( f(x) = 2x^2 - 10x + 4 \), substituting \( b = -10 \) and \( a = 2 \) into the formula, we get \( x = \frac{10}{4} = 2.5 \). Thus, the axis of symmetry is \( x = 2.5 \).
Mathematically, the axis of symmetry for a quadratic equation \( ax^2 + bx + c \) is given by the formula \( x = -\frac{b}{2a} \). This gives us the x-coordinate of the vertex. In the given function \( f(x) = 2x^2 - 10x + 4 \), substituting \( b = -10 \) and \( a = 2 \) into the formula, we get \( x = \frac{10}{4} = 2.5 \). Thus, the axis of symmetry is \( x = 2.5 \).
Minimum Value
The minimum value of a quadratic function that opens upwards is found at its vertex, where the axis of symmetry intersects the parabola. This is the lowest point on the graph. To find the minimum value, you substitute the x-value from the axis of symmetry back into the quadratic equation.
For \( f(x) = 2x^2 - 10x + 4 \), we determined that the axis of symmetry is \( x = 2.5 \). Plugging this into the function, we calculate: \( f(2.5) = 2(2.5)^2 - 10(2.5) + 4 \). First, compute \( (2.5)^2 = 6.25 \), then \( 2 \times 6.25 = 12.5 \). Subtract \( 25 \) (which is \( 10 \times 2.5 \)) and add 4, giving us \( 12.5 - 25 + 4 = -8.5 \). Hence, the minimum value is \(-8.5\).
For \( f(x) = 2x^2 - 10x + 4 \), we determined that the axis of symmetry is \( x = 2.5 \). Plugging this into the function, we calculate: \( f(2.5) = 2(2.5)^2 - 10(2.5) + 4 \). First, compute \( (2.5)^2 = 6.25 \), then \( 2 \times 6.25 = 12.5 \). Subtract \( 25 \) (which is \( 10 \times 2.5 \)) and add 4, giving us \( 12.5 - 25 + 4 = -8.5 \). Hence, the minimum value is \(-8.5\).
Quadratic Formula
The quadratic formula is a powerful tool used to find the roots or solutions of a quadratic equation. It is expressed as \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \). While the quadratic formula is not directly needed to find the vertex or minimum/maximum value of the quadratic function, it is crucial when solving for where the function equals zero (the x-intercepts).
This formula provides solutions for \( x \) by accurately computing the values where \( ax^2 + bx + c = 0 \). Understanding this concept can help you solve a wide range of quadratic problems, expanding your problem-solving toolkit when dealing with quadratic functions.
This formula provides solutions for \( x \) by accurately computing the values where \( ax^2 + bx + c = 0 \). Understanding this concept can help you solve a wide range of quadratic problems, expanding your problem-solving toolkit when dealing with quadratic functions.
Other exercises in this chapter
Problem 15
For the following exercises, find the \(x\) - or t-intercepts of the polynomial functions. $$ f(x)=x^{3}+x^{2}-4 x-4 $$
View solution Problem 15
For the following exercises, find the degree and leading coefficient for the given polynomial. $$ x\left(4-x^{2}\right)(2 x+1) $$
View solution Problem 16
For the following exercises, write an equation describing the relationship of the given variables. \(y\) varies jointly with \(x\) and \(z\) and when \(x=2\) an
View solution Problem 16
For the following exercises, find the inverse of the functions. $$ f(x)=\sqrt{2 x+1} $$
View solution