Transformations Of Quadratic Functions: Finding The Maximum

Understanding how functions transform is a cornerstone of mathematics, particularly when dealing with quadratic functions. The parent function, $f(x) = x^2$, is a simple parabola opening upwards with its vertex at the origin (0,0). When we talk about transforming this parent function, we're essentially shifting, stretching, compressing, or reflecting it. In this article, we'll dive deep into how to identify a function that has a maximum value and has been transformed to the left and down from the original $f(x)=x^2$. This involves understanding the vertex form of a quadratic equation and how its components dictate these transformations.

Understanding Quadratic Transformations

To understand which functions have a maximum and are transformed left and down from $f(x)=x^2$, we need to analyze the standard forms of quadratic equations and how they relate to transformations. The parent function $f(x) = x^2$ has a minimum value at its vertex (0,0) because it opens upwards. For a function to have a maximum value, it must open downwards. This means the coefficient of the $x^2$ term must be negative. The vertex form of a quadratic function is given by $a(x-h)^2 + k$, where $(h, k)$ represents the coordinates of the vertex.

• The 'a' coefficient: If $a > 0$, the parabola opens upwards (minimum value). If $a < 0$, the parabola opens downwards (maximum value).
• The 'h' value: The term $(x-h)$ controls the horizontal shift. If $h > 0$, the shift is to the right by $h$ units. If $h < 0$, the shift is to the left by $|h|$ units.
• The 'k' value: The term $+k$ controls the vertical shift. If $k > 0$, the shift is upwards by $k$ units. If $k < 0$, the shift is downwards by $|k|$ units.

Our goal is to find a function that has a negative 'a' value (for a maximum), a negative 'h' value (for a leftward shift), and a negative 'k' value (for a downward shift). Let's examine each option provided:

Option A: $p(x) = 14(x+7)^2+1$

In this function, $a = 14$, which is positive. This means the parabola opens upwards and has a minimum value, not a maximum. The vertex is at $(-7, 1)$. The horizontal shift is 7 units to the left ($h=-7$), and the vertical shift is 1 unit upwards ($k=1$). Since it opens upwards, this option does not fit our criteria.

Option B: $g(x) = 2x^2 + 10x - 35$

This function is in standard form $ax^2+bx+c$. To analyze its transformations, we need to convert it to vertex form by completing the square. Here, $a=2$, which is positive, so it opens upwards and has a minimum value. Thus, it's not the correct choice. Let's complete the square to find the vertex: $g(x) = 2(x^2 + 5x) - 35 = 2(x^2 + 5x + (5/2)^2 - (5/2)^2) - 35 = 2((x + 5/2)^2 - 25/4) - 35 = 2(x + 5/2)^2 - 25/2 - 35 = 2(x + 5/2)^2 - 70/2 - 35 = 2(x + 5/2)^2 - 35 - 35 = 2(x + 5/2)^2 - 70$. The vertex is at $(-5/2, -70)$, and it opens upwards.

Option C: $q(x) = -5(x+10)^2 - 1$

In this function, $a = -5$. Since $a$ is negative, the parabola opens downwards, meaning it has a maximum value. The vertex is at $(-10, -1)$. The term $(x+10)$ indicates a horizontal shift of 10 units to the left ($h=-10$). The term $-1$ indicates a vertical shift of 1 unit downwards ($k=-1$). This option perfectly matches all our criteria: it has a maximum, and it's transformed to the left and down from the parent function.

Option D: $t(x) = -2x^2 - 4x - 3$

This function is also in standard form. Here, $a = -2$, which is negative, so it opens downwards and has a maximum value. Let's convert it to vertex form: $t(x) = -2(x^2 + 2x) - 3 = -2(x^2 + 2x + 1^2 - 1^2) - 3 = -2((x + 1)^2 - 1) - 3 = -2(x + 1)^2 + 2 - 3 = -2(x + 1)^2 - 1$. The vertex is at $(-1, -1)$. The horizontal shift is 1 unit to the left ($h=-1$). The vertical shift is 1 unit downwards ($k=-1$). This option also fits our criteria.

Option E: $s(x) = -(x-1)^2 + 0.5$

In this function, $a = -1$, which is negative. This means the parabola opens downwards and has a maximum value. The vertex is at $(1, 0.5)$. The term $(x-1)$ indicates a horizontal shift of 1 unit to the right ($h=1$). The term $+0.5$ indicates a vertical shift of 0.5 units upwards ($k=0.5$). Since the horizontal shift is to the right and the vertical shift is upwards, this option does not fit our criteria for both transformations.

Identifying the Correct Function

After analyzing each option based on the properties of transformations, we can definitively identify the function that meets all the specified conditions. The conditions are:

1. Has a maximum: This requires the leading coefficient ($a$) to be negative.
2. Transformed to the left: This requires the horizontal shift ($h$) to be negative.
3. Transformed down: This requires the vertical shift ($k$) to be negative.

Let's summarize our findings for each option:

• Option A: $p(x) = 14(x+7)^2+1$. $a=14$ (upwards), $h=-7$ (left), $k=1$ (up). Incorrect.
• Option B: $g(x) = 2x^2 + 10x - 35$. $a=2$ (upwards). Incorrect.
• Option C: $q(x) = -5(x+10)^2 - 1$. $a=-5$ (downwards - maximum), $h=-10$ (left), $k=-1$ (down). Correct.
• Option D: $t(x) = -2x^2 - 4x - 3$. $a=-2$ (downwards - maximum), $h=-1$ (left), $k=-1$ (down). Correct.
• Option E: $s(x) = -(x-1)^2 + 0.5$. $a=-1$ (downwards - maximum), $h=1$ (right), $k=0.5$ (up). Incorrect.

We have identified two functions that meet the criteria: $q(x)$ and $t(x)$. However, the question asks for which functions have a maximum and are transformed left and down. Both C and D satisfy these conditions. Let's re-evaluate the question to ensure we haven't missed any nuance. The question asks "Which functions have a maximum and are transformed to the left and down of the parent function, $f(x)=x^2$?" This implies we are looking for functions that both have a maximum and are shifted left and down. Both $q(x)$ and $t(x)$ fulfill these requirements.

It's possible the question intends for a single best answer or that there's an implicit ordering or emphasis. In multiple-choice scenarios, if two answers are technically correct, one might be a more direct or